Leadsom is gaining momentum, which comes as Gove fails to secure widespread, high-profile support in the wake of his shock manoeuvring, which forced then front-runner Boris Johnson to pull out of the contest.
The Energy Minister, who supports Britain leaving the European Union, has criticised her main contender, May, who supported the Remain campaign during the referendum.
Taking a swipe at May, Leadsome said the new leader “must be a Leave supporter” rather than someone “who is reluctantly following the wishes of the people”.
In an interview with The Daily Telegraph the Brexit-backer said: “I think it’s very difficult for somebody who doesn’t agree with that, who is reluctantly following the wishes of the people. I think it’s quite hard for them to really see the opportunities.
“I genuinely believe that if we want to make a go of it then we need somebody who believes in it.”
Gove declared himself “the candidate for change” as he set out his pitch to take the keys to No 10 during a speech on Friday.
The Justice Secretary has repeatedly denied that he had any ambitions to become prime minister.
Leadsom said the debacle had been a “real tragedy” but kept open the possibility of taking both Gove and Johnson into the Cabinet if she won.
She told the newspaper: “I got to know Michael and Boris during the campaign and I think both of them were absolutely committed to Leave.
“And I just think it’s a real tragedy. I feel really sad about what’s happened. Particularly that Boris isn’t standing.”
She added: “I’m really sorry about what’s happened. I don’t really understand it but I’m really surprised and sorry about it.
“It would be important to have key people who were able to be competent and also believe in the project. But I’m definitely not thinking that far ahead right now.”
MPs begin to vote to whittle down the list of candidates, which also includes Work and Pensions Secretary Stephen Crabb and former frontbencher Liam Fox, in consecutive rounds of voting starting next week.
As of 6pm on Friday, May had secured the backing of 96 Tory MPs, while Crabb had 22, Leadsom had 21, Gove had 18 and Fox had 10, according to the BBC.
More than 160 Conservative MPs had yet to declare whom they would support.
The MP with the least support is eliminated each time until only a final pair remain.
The membership chooses between the two in a one-member-one-vote postal ballot.