Previously, ministers had avoided talking about a job that did not exist
Tony Blair will be the UK's official candidate for EU president, Baroness Kinnock has apparently confirmed.
The post will only be created if the Lisbon Treaty is ratified by all EU states - Ireland is to hold a second referendum in October.
There has long been speculation that the former PM would go for the post but he has not confirmed he will do so.
It is the first time a UK government minister has publicly announced Tony Blair is a candidate for the job.
Previously, ministers have said it was premature to talk of supporting any particular candidate when the job itself did not exist.
'Strength of character'
But at a briefing in Strasbourg, Europe Minister Lady Kinnock said: "The UK government is supporting Tony Blair's candidature for president of the Council [of EU governments]."
Asked if it had been discussed with Mr Blair, currently a Middle East envoy, she said: "It is the government's position. I am sure they would not do that without asking him."
Lady Kinnock went on to say Mr Blair had the "strength of character" and "status" to take on the job.
She added: "People know who he is, and he could step into this new role with a lot of respect and he would be generally welcomed."
But there have been reports that Mr Blair's role in the Iraq war and Britain's failure to join the euro, could go against him.
Mr Blair's spokesman said later: "As we have said, time and again on this, there is nothing to be a candidate for since the job doesn't actually exist."
Ireland was the only EU member state to hold a public vote on the treaty, which must be ratified by all 27 countries. The treaty was rejected in that referendum last June. It will hold a second referendum in the autumn.
The precise role of the EU president has not been laid out. The Swedish government, which currently holds the six-month EU presidency, has suggested drawing up a proper job description before seeing which available political figure fits it best.
Downing Street said if Mr Blair wants to stand for the job, the government would support him but it was up to him to decide if he wanted to put his name forward.
The prime minister's spokesman declined to say whether Mr Brown had discussed the post with Tony Blair or had encouraged him to stand.
The Conservatives, whose efforts to get a UK referendum on the treaty were defeated in the Commons, said the British people had not been given a say on whether Europe should have a new president.
Shadow foreign secretary William Hague said the new role could be "enormously damaging".
"Any holder is likely to try to centralise power for themselves in Brussels and dominate national foreign policies.
"In the hands of an operator as ambitious as Tony Blair, that is a near certainty. He should be let nowhere near the job."
Former Spanish prime minister Felipe Gonzalez, a socialist, is seen as a potential rival to Mr Blair for the presidency job.
According to Jean-Marie Colombani, a former editor of the French daily Le Monde, French President Nicolas Sarkozy now favours Mr Gonzalez for the job. Earlier Mr Sarkozy had been seen as a powerful Blair supporter.
Mr Gonzalez, 67, currently heads a group of "wise men" - leading experts from various backgrounds - appointed to help shape the EU's future.
But Italian Prime Minister Silvio Berlusconi has described Mr Blair - a longstanding friend - as the "ideal personality" to be EU president.
Another potential rival is Jean-Claude Juncker, the veteran centre-right Luxembourg prime minister who has been at the heart of key EU negotiations since 1992.
He has said the EU presidency job would interest him if it has sufficient clout.
Campaign group Open Europe said the EU should stick with the rotating presidency which allows the current national leaders - who have a democratic mandate - to set the EU's agenda on a six-monthly basis.
Director Lorraine Mullally said: "Tony Blair may well be a respected politician around the world, but he is also yesterday's news.
"The problem with creating a post of EU president is that it would inevitably go to an ex-leader who has lost the support of his or her own electorate."