UK Foreign Secretary David Miliband has joked that the European Union might "not be able to afford" Tony Blair as its first president.
Mr Brown is expected to back Mr Blair for the job
Mr Miliband's comment on BBC Radio 2's Jeremy Vine show came in the wake of Mr Blair picking up high-paid City roles.
But he said: "Let's get the job figured out first, then let's go on and figure out who the right person is."
He added that he would "be happy" if Mr Blair did the job, and said he thought Gordon Brown would also be happy.
The foreign secretary was replying to Lib Dem leader Nick Clegg's suggestion that an attempt by Tony Blair to become President of the European Council would be an "act of political vanity".
Mr Blair has discussed the idea with Gordon Brown, his successor as prime minister, with senior Downing Street figures urging him to launch a bid.
But Mr Clegg said backing Mr Blair would be "an embarrassing diversion".
The role of EU Council president - dubbed by many as "president of Europe" - will come into being next year, if the organisation's 27 member states ratify the Lisbon Treaty, signed by national leaders in December.
The exact nature and status of the role is yet to be decided.
Some countries want a high-profile figure to represent the EU on the world stage, while others propose a more low-key, bureaucratic figure.
French president Nicolas Sarkozy has openly backed Mr Blair.
In a speech last month, seen by many as the launch of a possible EU presidency bid, Mr Blair called for "new ideas" on Europe.
The other main figure being mentioned for the job - which will be filled by majority voting - is Luxembourg's prime minister Jean-Claude Juncker.
German chancellor Angela Merkel is also thought to be a possible candidate, if she loses her country's general election next year.
Under the terms of the Lisbon Treaty, a politician would be chosen to be president of the European Council for two-and-a-half years, replacing the current system where countries take turns at being president for six months.
Mr Blair stood down as prime minister last June after more than 10 years in power.
He is currently working as a peace envoy to the Middle East and has taken up advisory roles with insurance firm Zurich and investment bank JP Morgan - and is reported to have signed a £5m-plus deal for his memoirs.