Andrew Grice: Unwelcome shadows lurking behind Brown

Behind the scenes, two other names are on many lips in Euroland: Tony Blair and David Cameron

Gordon Brown was centre-stage at the European Union summit which ended yesterday, as he lobbied hard for a ground-breaking agreement to combat the economic crisis, ahead of the G20 meeting in London in 12 days. "It is important we achieve a united front, finding common solutions to common problems," he told everyone he could find.

Yet behind the scenes, two other British names are on many lips in Euroland: David Cameron and Tony Blair. Leaders and officials study the opinion polls in other countries as well as their own. While most admire Mr Brown's Herculean efforts to get his "global new deal", they expect Mr Cameron to be attending EU gatherings from June next year. Mr Blair, meanwhile, is being tipped for the new post of "President of Europe", representing the EU on the world stage and driving through the agenda set by its 27 leaders.

Opinions of Mr Cameron differ sharply on the Continent. For some, he would head the most Eurosceptic British Government ever – and they still bear the bruises from Margaret Thatcher's handbag.

Mr Cameron would demand Britain's withdrawal from the EU social chapter of workers' rights.

That would swiftly be rejected by all 26 other member states. He might then try to renegotiate Britain's terms of EU membership, which could be a slippery slope towards the exit door (a direction favoured secretly by a minority of Tories). There is bemusement here in Brussels about Mr Cameron's decision to take his party's MEPs out of the mainstream centre-right group in the European Parliament, the European People's Party (EPP), after elections to it in June. It is too federalist for his tastes, and he believes it is dishonest to say "one thing in London and another in Brussels".

So the supposedly modernised Tories will walk away from the parties of Nicolas Sarkozy and Angela Merkel to form a new gang with a motley collection of Eurosceptics. The only two members signed up so far include an anti-gay rights Polish party, one of whose MPs warned that a Barack Obama victory would mean "the end of the civilisation of the white man", and a Czech party whose founder describes climate change as a "global myth". It's a bit odd for Mr Cameron to expend so much energy on erasing the Tories' "nasty party" image at home, only to join the nasties in Europe.

Does it matter? Europe is not a big deal for most British voters. It is an issue Mr Cameron has played down since becoming Tory leader, admitting that "banging on" about Europe turned off the voters. But he has had to make promises – on the social chapter and the EPP – to keep his Eurosceptic-dominated MPs happy.

David Davis, his rival for the Tory leadership in 2005, reckons the EPP pledge won Mr Cameron the votes of 40 MPs (one in five).

Europe matters because it sends a signal to the voters. Some of them like Mr Cameron but are not so sure about his party. They don't want it to revert to type. The British public may not love Europe but they don't want to pull out of the EU. The financial crisis has illustrated the need for governments to act together. Mr Cameron accepts the need for EU co-operation on climate change. Recessions, too, do not respect borders.

Some senior Brussels figures who have held talks with Mr Cameron suspect his bite would be less strong than his bark, that he would prove a constructive European, realising that it would be in Britain's interests to cut deals and make compromises. "You cannot be a vegetarian in a beefeater club," one said this week.

Close allies predict Mr Cameron will concentrate on the economic problems he will inherit and let sleeping European dogs lie. But Europe could still rise up and bite him.

He has placed the two most Eurosceptic members of the Shadow Cabinet, William Hague (foreign affairs) and Liam Fox (defence) in the posts that would deal most with the EU in government. Tory Europhobes have been keeping their heads down because they are convinced Mr Cameron is "one of us". Their ranks would be bolstered by a new intake of Tory MPs, especially if the party wins big. If he acts soft on Europe after becoming prime minister, they could make his life very difficult. It happened to John Major. The hardliners thought he was "one of us" too.

Mr Blair is the talk of the town again in Brussels because hopes are rising that Ireland will vote "yes" to the Lisbon Treaty in a second referendum this autumn. The treaty would create the "President of Europe" job, replacing the ineffective "musical chairs" system under which one country holds the EU's rotating chairmanship for six months.

Some EU insiders believe a Brit is unlikely to land the president's post because the UK is outside the EU's inner core of the euro and the Schengen "open borders" agreement.

But there is also a growing recognition that if the EU wants clout on the global stage, it will need a heavy-hitter and a big name. Mr Blair foots the bill. Opposition to him because of his hawkish stance in Iraq, which included some of Labour's socialist allies, has faded with time. Others may be in the running – including Ms Merkel, if she loses power in September. But Mr Blair's name seems to be in the frame. So Prime Minister Cameron, who once called himself the "heir to Blair" and whose style often recalls the man on which he he modelled himself, may find himself negotiating the terms of Britain's EU membership with President Blair. It could be a quite a match: Blair Mark One versus Blair Mark Two.