The comments came after British media quoted Europe Minister Glenys Kinnock as saying Blair was expected to seek the post.
The new role is due to be created as part of reforms under the Lisbon Treaty, which has yet to come into force, largely due to public resistance in Ireland.
The treaty, which aims to give the 27-member bloc stronger leadership, fairer decision-making and more of a say on the world stage, could be ratified later this year.
"It's the prime minister's view that Tony Blair would be a good candidate for any big international job," Prime Minister Gordon Brown's spokesman told reporters.
"If Tony Blair decides to stand as president of the European Council, once that job has been created, then of course we will support him."
There has been long-standing speculation that Blair would seek the role. It resurfaced again on Wednesday after comments made by Europe Minister Kinnock in Strasbourg.
"The UK government is supporting Tony Blair's candidature for president of the Council," the BBC quoted her as saying at a briefing. "It is the government's position. I am sure they would not do that without asking him."
Blair stood down as prime minister in 2007 after ten years in power and has since taken up the role of Middle East envoy, representing the quartet of mediators made up of the United States, the European Union, the United Nations and Russia.
The relationship between Blair and Brown, the architects of Labour's first decade in government, was infamously rocky for much of the former's time in charge, with both men tussling for influence over the party and the direction of policy.
Britain's main opposition Conservatives, who are tipped to win an election which must be called by mid-2010, want the European Union to have less power over its member states and oppose the creation of the role of European president.
"Any holder is likely to centralise power for themselves in Brussels and dominate national foreign policies," said Conservative foreign affairs spokesman William Hague. "In the hands of an operator as ambitious as Tony Blair, that is a near certainty."
Blair's office denied that he was preparing to stand for the European Council president role, which would be appointed by EU leaders on a renewable two-and-a-half year term.
"There is no campaign," Blair's spokesman said. "As we have said time and again on this, there is nothing to be a candidate for since the job doesn't actually exist."
(EurActiv with Reuters.)
