Speaking earlier this week, Mrs Merkel said the EU needed more powers to ensure budget discipline among member states and Germany was willing to give up sovereignty in some areas to facilitate this.
In contrast, Mr Cameron is pushing for the UK to take powers back from Brussels in pursuit of what he says is his goal of a more flexible and diverse Europe.
In the event of eurozone members moving closer together, the UK has said it wants safeguards that those outside the single currency area will not be disadvantaged in terms of access to the single market and regulations on key sectors like finance.
But there are concerns Germany is determined to push ahead with a tax on financial transactions, known as the Tobin Tax, in an effort to regulate the markets.
The idea, also backed by French President Nicolas Sarkozy, has caused alarm in the UK amid concerns that the US, China and other major economies may not come on board and the City of London would be seriously damaged if the tax only applied across Europe.
British ministers reacted angrily to recent remarks by a senior member of Mrs Merkel's party, Volker Kauder, who suggested the UK may need to make concessions on the issue of the tax and must show more collective responsibility in Europe.
Chancellor George Osborne said a Europe-only tax would be "economic suicide" while Business Secretary Vince Cable said the UK would not "fall for" such an idea, which he said had nothing to do with the urgent problems facing Europe on productivity and employment.
Ahead of Friday's meeting, Labour's Chris Bryant - a former Europe minister in the last government - said the coalition was "not managing to get its argument across" and was instead making "injudicious" comments about other countries' economies.
"I think we should have been doing much more to bind together a caucus of Britain, France, Germany to provide much more leadership on this issue," he told BBC One's Question Time.
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