Ken Clarke has said he is waiting to see if the Conservatives are "leadable" before deciding whether to enter the race to succeed Michael Howard.
Mr Clarke has run for the leadership on two previous occasions
He said he wanted to know if the Tories were prepared to be a socially liberal, centre-right party based around a belief in the free market.
His comments come amid a row over the timing of Mr Howard's exit.
He has been urged to go early so his successor can launch their leadership bid at the party's autumn conference.
Cameron to run?
But Mr Howard wants to reform the party's leadership election rules before he steps down in time for Christmas and he has dismissed calls for an earlier resignation.
Mr Clarke, whose pro-EU views have put him at odds with many in the Conservative Party, indicated a French 'No' vote in their referendum on the European Constitution could well help him.
"It would make things a lot easier for the Conservative Party if it knows Europe is on the backburner," he said adding that a French 'No' vote would "kill" the issue.
Tory education spokesman David Cameron, himself seen as a possible contender in the leadership election, meanwhile defended Mr Howard saying he had done "an excellent job".
"Everyone agreed after the election that it would be a good idea to have Michael stay on as leader while we look at why we lost the election, look at the issues and review our policies," said Mr Cameron.
Rumoured to be Mr Howard's favourite for next Tory leader, Mr Cameron would not state a preference over who he was planning to back in the contest.
Ray Monbiot, who chairs the party's national convention, said existing rules meant a swift election was just not possible.
"It is not just a question of snapping the fingers and saying 'OK, we will get a new leader within two or three weeks', because the mechanism isn't there to do it," he told BBC News.
Current rules put in place under William Hague and then used after he lost the 2001 election saw a summer-long contest that ended with the election of Iain Duncan Smith after the party's rank and file backed him over Mr Clarke - the MPs' preferred choice.
On Thursday Mr Howard wrote to MPs admitting plans for reforming the party "could have been better handled".
But he said it would not be in the party's interests to plunge it into an immediate leadership election.
A major contributor to the Tory party, Sir Tom Cowie, told the BBC Mr Howard should go "in hours or days".
Some Tory MPs have said they are angry that proposals to reform the party and change how its leader is elected have been presented as a fait accompli.
Howard: Sticking to timetable
And there is talk of signatures being gathered at Westminster to trigger an early leadership contest.
The proposed new leadership system would return the final decision to MPs, with grassroots members showing their views in a vote of local Tory association chairmen.