A former Italian premier, a respected politician and a southern European - it's their "turn" for the job - Mr Prodi seems ideally placed. He has been endorsed by his successor as prime minister, Massimo D'Alema, and praised by Tony Blair who is a known fan. The two met last summer when the Blair family were holidaying in Tuscany, then shared a platform at a Washington summit on the "third way".
Mr Prodi is available immediately and, if he gets the job, Italy may be more willing to increase its payments into EU coffers, helping Germany to reduce its pounds 8bn annual net contributions.
But a couple of factors count against the Italian. First he is not a socialist (again it's their "turn" for the job) and 11 of the 15 heads of government belong to that grouping; and second he is the clear front- runner. Historically, the obvious choice is rarely the successful one.
Predicting the outcome of negotiations is risky because, technically speaking, the vote of Luxemburg's leader is as important as that of Germany's. Moreover, other negotiations tend to become intertwined in the horse-trading.
Unlike the last time this contest came around, in 1994, the field includes a number of strong potential candidates, including the Spaniard, Javier Solana, Secretary General of Nato. He has not been proposed by his prime minister, Jose Maria Aznar, who comes from a rival party, but Mr Aznar would not block the appointment if others are keen. Mr Solana's allies believe he could extract himself from his current job, despite the Kosovo crisis, after next month's Nato summit in Washington.
Then there are the undeclared contenders. These include Wim Kok, the Dutch premier and a close ally of Mr Blair. Mr Kok proved how close he is to the Labour leader by endorsing him before he became prime minister, putting John Major's nose out of joint. But as prime minister of the Netherlands he would only allow his name to go forward if he was reasonably sure of success, for fear of damaging his domestic political position.
The same is true of Antonio Guterres, the Portuguese premier, whose socialist party at home cannot afford to do without him in elections due this autumn. A delay in the decision-taking, with a caretaker filling in, would suit him. Other possible contenders include the German defence minister, Rudolf Scharping, and even the man vetoed by John Major in 1994 - Belgium's premier Jean-Luc Deheane.
Then there are the caretaker candidates. Sir Leon Brittan, former home secretary and now a vice president of the Commission, clearly wants that role. His main problem is his nationality; with Britain outside the euro it is in a weak position to put up a candidate. Karel Van Miert, the senior Belgian commissioner, is another contender.
The caretaker scenario arises because, if a new president is appointed quickly, they would have to be vetted by the European Parliament twice: first by the current assembly, then by the new parliament which will be elected in June.
Which brings us back to Mr Prodi, who does not want the caretaker role, but might be willing to jump through the hoops twice in nine months. That would mean taking over immediately on a temporary basis, providing he has the political backing to perform the job for the full 2000-2005 term.
On Friday the former Italian premier had some mixed news. He received the sort of endorsement some believe he could do without: that of the outgoing president, the luckless Mr Santer.
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