Blair, who led Britain into the Iraq war, remains a controversial figure for many, but his supporters argue that he is one of the few people with the stature necessary for the job.
His opponents, meanwhile, point to what they see as Blair's profound unsuitability for the role, viewing him as someone who consistently kept at a distance from Europe, evidenced by acts such as securing an exemption for Britain from the European Charter of Fundamental Rights.
The Benelux countries have sought in recent days to block Blair from getting the EU top job. Britain is regarded by Belgium, Luxembourg and the Netherlands as too Eurosceptic and not sufficiently pro-EU to take the new job, which is created by the Lisbon Treaty.
According to the British press, a confidential paper which does not mention Mr Blair by name has been circulated by the three countries. "The president must have the stature of a head of state or government. He must be someone who has demonstrated his commitment to the European project," the paper says.
Blair's critics cite the example of the recent appointment of new NATO Secretary-General Anders Fogh Rasmussen. A high ranking EU diplomat told EurActiv that Rasmussen's stewardship of the military alliance is "such a mess", arguing that EU countries should look for team players to fill in the top jobs of permanent Council president and High Representative for foreign affairs and security issues.
Rasmussen, a former long-serving Danish prime minister, has repeatedly failed to consult member states on major issues, the diplomat said, which runs against well-established traditions and is unacceptable to many NATO members. One example is a recent landmark speech in which Rasmussen apparently recognised Russia's sphere of influence, the diplomat said (EurActiv 21/09/09).
In recent weeks, Luxembourg has sought to rally opposition to Blair, describing the former UK prime minister as a divisive figure. Blair "doesn't have the right stature on EU issues or on the big issues of world politics," Luxembourg Foreign Minister Jean Asselborn told the Sueddeutsche Zeitung in an interview published earlier this month. "He has more often divided than united."
In Britain, the reactivation of the Stop Blair! petition gained hundreds of new signatures in just a few days, with some 37,000 signatures now attached to the petition.
Blair violated international law when he committed his country to a war in Iraq that a large majority of European citizens opposed, underline his critics. "His role in the Iraq war would weigh heavily on the image of the Union in the world, should he in fact be named its president," according to advocates of the Stop Blair! campaign.
Rather than move European integration forward, the former UK prime minister set a series of so-called red lines during the Lisbon negotiations, with the intent of blocking any progress on social issues and tax harmonisation, as well as common defence and foreign policy, the campaigners stressed.
Daniel Cohn-Bendit, joint leader of the Greens in the European Parliament, agreed, saying "there is no way Blair should get this job. He did nothing to defend European interests when he was PM and has done even less as a Middle East envoy".
Meanwhile, Blair finds himself faced with a new hurdle after recent media reports claimed that he accepted tens of thousands of pounds from a steel billionaire campaigning for Ukraine to join the European Union.
Victor Pinchuk, who is championing the country's bid for EU membership, has already hired Stephen Byers, the former Labour cabinet minister, to press his case. He reportedly paid Blair to give a keynote speech in Ukraine.
Several European countries support closer economic ties with Ukraine but have not directly backed an EU membership bid. Some, including Germany, are concerned that closer links with Ukraine may jeopardise relations with Russia.
However, despite the growing chorus of opposition, Irish gambling company Paddy Power - a firm well known for weighing into political speculation - has made Blair its odds-on favourite to take the job.
Other names floated for the Council president post include Dutch Prime Minister Jan Peter Balkenende (EurActiv 29/09/09) and Luxembourg's Jean-Claude Juncker.