French President Nicolas Sarkozy, who was originally Blair's key backer, is said to have 'changed his mind'. The new favourite is Spain's charismatic socialist former prime minister Felipe Gonzales, The Sunday Telegraph reports.
European leaders are just beginning the complex horse-trading that will end in the appointment of a EU president and 'high representative' - effectively the union's foreign minister - once the Lisbon Treaty has been finally fully ratified.
That could come in October when Ireland stages its second referendum on the treaty following a 'No' vote last year. Pundits expect a 'Yes' vote this time round which would allow the treaty to pass into law among the EU's 27 member states.
Blair, currently a Middle East peace envoy, remains 10 Downing Street's choice. A Foreign Office source told The Sunday Telegraph, "The new job is perfectly suited to a big personality like Tony Blair."
Sarkozy is thought to have altered his stance - partly because of Blair's perceived "failure" to achieve any change in the Middle East. He has now switched support to Gonzales, 67, who has claimed that Europe has been "adrift for 20 years" and has become a mere follower of the United States.