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She told the Bundestag that a new EU treaty was needed to set up such a union and impose budget discipline.
On Monday she is to meet French President Nicolas Sarkozy, who has also called for EU treaty changes.
EU leaders have been under pressure to do more to tackle the debt crisis, amid concern about the survival of the euro.
In her speech, Mrs Merkel promised "concrete steps towards a fiscal union" - in effect close integration of the tax-and-spend polices of individual eurozone countries, with Brussels imposing penalties on members that break the rules.
"We need budget discipline and effective crisis management mechanism," she said. "So we need to change the treaties or create new treaties."
The German government has been pressing for changes to establish powers to veto national budgets in the eurozone that breach agreed rules.
"We have started a new phase in European integration," Mrs Merkel said.
On Thursday, Mr Sarkozy said a new European treaty governing relations between members was necessary to protect Europe's place in the world.
"We must confront those who doubt the stability of the euro and speculate on its break-up with total solidarity," Mr Sarkozy said.
Mr Sarkozy said the euro could not continue to exist unless eurozone economies pulled together, with France and Germany playing a key role to ensure "a zone of stability".
However Mr Sarkozy rejected suggestions that national budgets could be approved and regulated in Brussels, and said France would not give up its sovereignty.
No 'joint liability'
In her speech on Friday, Mrs Merkel also reiterated her opposition to the European Central Bank (ECB) issuing "eurobonds" backed by all eurozone members.
"A joint liability for others' debts is not acceptable," she said. "Eurobonds are not a rescue measure in this crisis."
She added: "National responsibility and European solidarity serve each other."
Many in Germany argue that eurobonds would penalise countries with a high credit rating, and reduce the incentive of indebted governments to reform.
Continue reading the main story
A default can have a number of important implications. If a borrower is in default on any one debt, then all of its lenders may be able to demand that the borrower immediately repay them. Lenders may also be required to write off their losses on the loans they have made.
However Germany's Sueddeutsche Zeitung daily said Mrs Merkel was willing to see the ECB step up its buying of bonds from indebted eurozone countries, as a bridging solution until budget controls took hold.
During their meeting in Paris on Monday, Mr Sarkozy and Mrs Merkel are to agree on joint proposals to be put to a meeting of European leaders next week.
Many analysts see that summit as a crucial moment in efforts to tackle the debt crisis.
European stock markets rose early on Friday, as Europe's leaders call for closer economic integration as the way to resolve the debt crisis
Meanwhile Mr Sarkozy is discussing the eurozone crisis with UK Prime Minister David Cameron in Paris.
Britain is concerned about the possible impact of a two-speed Europe, in which it could be left on the margins along with other countries outside the euro.
Mr Sarkozy and Mrs Merkel have been trying to reassure markets about the single currency
German Chancellor Angela Merkel has said Europe is working towards setting up a "fiscal union", in a bid to resolve the eurozone's debt crisis.
On Monday she is to meet French President Nicolas Sarkozy, who has also called for EU treaty changes.
EU leaders have been under pressure to do more to tackle the debt crisis, amid concern about the survival of the euro.
In her speech, Mrs Merkel promised "concrete steps towards a fiscal union" - in effect close integration of the tax-and-spend polices of individual eurozone countries, with Brussels imposing penalties on members that break the rules.
"We need budget discipline and effective crisis management mechanism," she said. "So we need to change the treaties or create new treaties."
The German government has been pressing for changes to establish powers to veto national budgets in the eurozone that breach agreed rules.
"We have started a new phase in European integration," Mrs Merkel said.
Continue reading the main story 
Seven days of intense diplomacy have begun, before a EU summit next week. Already officials are calling it the "make or break" summit.
So is there a plan? First there is agreement between France and Germany that the flaws in the single currency are systemic. They require a major redesign of institutions and agreements.
Secondly, there is an acceptance that the eurozone is moving beyond a monetary union to a fiscal union, with much greater central control over tax and spending.
How far will this go? We do not know. Certainly there will be much closer supervision of national budgets with a structure to enforce rules on spending.
It is also unclear whether the French and Germans are considering a limited treaty change or something more substantial. Chancellor Merkel has spoken of "sinners" being dragged before the European Court of Justice. Enforcement remains a potential area of disagreement.
Analysis
So is there a plan? First there is agreement between France and Germany that the flaws in the single currency are systemic. They require a major redesign of institutions and agreements.
Secondly, there is an acceptance that the eurozone is moving beyond a monetary union to a fiscal union, with much greater central control over tax and spending.
How far will this go? We do not know. Certainly there will be much closer supervision of national budgets with a structure to enforce rules on spending.
It is also unclear whether the French and Germans are considering a limited treaty change or something more substantial. Chancellor Merkel has spoken of "sinners" being dragged before the European Court of Justice. Enforcement remains a potential area of disagreement.
But she made it clear that this was a long-term process that would take years.
"We must confront those who doubt the stability of the euro and speculate on its break-up with total solidarity," Mr Sarkozy said.
Mr Sarkozy said the euro could not continue to exist unless eurozone economies pulled together, with France and Germany playing a key role to ensure "a zone of stability".
However Mr Sarkozy rejected suggestions that national budgets could be approved and regulated in Brussels, and said France would not give up its sovereignty.
No 'joint liability'
In her speech on Friday, Mrs Merkel also reiterated her opposition to the European Central Bank (ECB) issuing "eurobonds" backed by all eurozone members.
"A joint liability for others' debts is not acceptable," she said. "Eurobonds are not a rescue measure in this crisis."
She added: "National responsibility and European solidarity serve each other."
Many in Germany argue that eurobonds would penalise countries with a high credit rating, and reduce the incentive of indebted governments to reform.
Continue reading the main story
Crisis jargon buster
Use the dropdown for easy-to-understand explanations of key financial terms:
Default
Strictly speaking, a default occurs when a borrower has broken the terms of a loan or other debt, for example if a borrower misses a payment. The term is also loosely used to mean any situation that makes clear that a borrower can no longer repay its debts in full, such as bankruptcy or a debt restructuring.
A default can have a number of important implications. If a borrower is in default on any one debt, then all of its lenders may be able to demand that the borrower immediately repay them. Lenders may also be required to write off their losses on the loans they have made.
Analysts also say Berlin is opposed to wholesale intervention by the ECB because of the country's experience of hyperinflation in the 1920s.
During their meeting in Paris on Monday, Mr Sarkozy and Mrs Merkel are to agree on joint proposals to be put to a meeting of European leaders next week.
Many analysts see that summit as a crucial moment in efforts to tackle the debt crisis.
European stock markets rose early on Friday, as Europe's leaders call for closer economic integration as the way to resolve the debt crisis
Meanwhile Mr Sarkozy is discussing the eurozone crisis with UK Prime Minister David Cameron in Paris.
Britain is concerned about the possible impact of a two-speed Europe, in which it could be left on the margins along with other countries outside the euro.
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