An attempt by Tony Blair to become President of the European Council would be an "act of political vanity", Lib Dem leader Nick Clegg has said.
Mr Brown is expected to back Mr Blair for the job
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.
BBC Europe editor Mark Mardell said Mr Brown was "waiting for a nod" from his predecessor before voicing his support.
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 BBC's Mardell said the former prime minister was "interested, but cautious" and only wanted the "job on his terms if the president is really going to be a strong presence" globally.
He added that, if he launched a campaign, Mr Brown would feel compelled to support him.
However, Mr Clegg, a former member of the European Parliament, said: "Tony Blair's candidacy for the post of president of the European Council is a display of political vanity and will not succeed.
"His foreign policy ended in disaster in Iraq, he divided Europe and wrenched Britain away from key allies.
"The EU presidency will not be powerful, but it will require a figure who can lead by consensus, not division.
"Gordon Brown should recognise that talk of 'President Blair is an embarrassing distraction for Britain in Europe and refuse to support him."
To become president, Mr Blair would need the backing of the majority of EU member states' governments.
The other main figure being mentioned for the job is Luxembourg's prime minister Jean-Claude Juncker.
'Good for Britain'
German chancellor Angela Merkel is also thought to be a possible candidate, if she loses her country's general election next year.
Labour MP and former Europe minister Denis MacShane told BBC Radio 4's Today programme: "I think he [Mr Blair] has been one of the biggest national leaders Europe has had in the last generation."
He added: "If he wants the job... from a British point of view it's always good to have a Brit in a top job."
And Commons leader Harriet Harman said it was always "good for Britain and good for Europe" when a leading British politician played a leading role in the EU.
An online petition against Mr Blair becoming president, organised by the European Tribune group, has attracted more than 2,900 petitions.
Conservative MP and former foreign secretary Sir Malcolm Rifkind said: "By his own standards, his [Mr Blair's] whole strategy on Europe failed miserably.
"He wants to be leader of Europe, but he was more responsible than anybody for dividing Europe over the Iraq war.
"He failed to get Britain into the single currency, I'm delighted to say, but he didn't even dare have a referendum...
"He didn't even ask the British public. He just dropped the idea. I would be astonished if the rest of Europe sees him as their natural spokesman."
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.