LONDON, June 24 (Reuters) - Some leading members of Britain's ruling Conservative Party want finance minister Philip Hammond to become prime minister, replacing Theresa May and steering the country through the Brexit process, the Sunday Times reported.

May, prime minister for less than a year, has been badly damaged by her failure to win a parliamentary majority in a June 8 election and is still working to persuade a small northern Irish party to support her Conservatives in government.

Hammond was sidelined for months by May during the election campaign and had been widely expected to be fired after the vote. But the party's poor showing has propelled him back into the spotlight.

Citing some members of May's top team, or cabinet, the Sunday Times said Hammond should be appointed as a caretaker prime minister to see the country through until 2019 when it officially leaves the European Union.

Brexit minister David Davis should be named as his deputy, the newspaper quoted sources as saying.

"I think Philip is the only plausible candidate for a couple of years, with DD (David Davis) running Brexit," the paper quoted a serving minister as saying.

"He is a more credible caretaker than the current prime minister. The PM's brand is so damaged it is painful. The calculation that people are beginning to make is that she is so inadequate we can't wait two years with her in place."

A former cabinet colleague was quoted by the paper as saying that Hammond believed he could do the job. Not all cabinet members were in agreement however, with some backing Davis and others favouring Foreign Secretary Boris Johnson.

A spokesman for May's Downing Street office declined to comment.

(Reporting by Kate Holton; Editing by Nick Zieminski)

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