David Davis, who has topped the first leadership poll of Tory MPs, says it is "extremely unlikely" he will lose supporters ahead of the next round.
He said he was "very grateful" for gaining 62 votes, even though it was fewer than the number of declared backers ahead of the ballot.
Mr Davis said he was still "first choice to go to the country" among his Conservative colleagues.
And he brushed off suggestions it was difficult to reverse a fall in support.
David Davis - 62
David Cameron - 56
Liam Fox - 42
Ken Clarke - 38
He said there would be a "lot of movement in votes" in the second round of the contest, which takes place on Thursday, and he would not rule out an increase in the number of MPs backing him.
He said he expected to gain some votes from Liam Fox, who finished third in the first round, but admitted it was "highly unlikely" any of David Cameron's supporters would back him.
Two candidates will go forward to a final ballot of the party membership, with the new leader expected to be crowned in early December.