Energy Secretary Amber Rudd, said in a Channel 4 News interview that while Leadsom is "a very good junior energy minister, but hasn't anywhere near the sort of experience necessary".
Conservative MPs have chosen Home Secretary Theresa May, and energy and climate change minister Andrea Leadsom as their final two candidates.
The party has said it plans to announce the new leader on September 9, giving the candidates two months to actively campaign for members' support. "This vote shows that the Conservative Party can come together, and under my leadership it will", Mrs May told supporters after the results were announced. The victor will replace Prime Minister David Cameron, who announced his resignation after Britain voted last month to leave the European Union.
Mr Gove has previously said that the top job would have to got to someone who campaigned to leave the European Union - a statement that would appear to rule out Mrs May, who backed a Remain vote.
"Those are the things that my colleagues have voted for in such numbers today and I am confident that those are things that will win the support of members across the country, and indeed the whole country".
Leadsom had also declined to release her taxes but now says she will, since she is in the final two.
Although Mrs May signed up to the Remain side, she maintained a low profile during the referendum campaign. He added: 'We need someone who can step into the PM's shoes and have the credibility to get on with the job right away'.
Conservative lawmakers in Britain are voting to decide which two candidates will face a runoff to be the country's next prime minister.
Mrs May has the backing of Guy Opperman, the Conservative MP for Hexham, and has now won the support of Mrs Trevelyan, MP for Berwick.
Ms Davidson told the BBC's Newsnight programme that Mrs May had the "vast majority" of the parliamentary party behind her, and had "huge support" in all parts of the United Kingdom, including among the Scottish Conservatives.
Speaking to reporters outside Parliament he said: "I believe Andrea now, going to the country, has all that fantastic combination of steel, real grit, real determination, real world experience - working 25 years out of this place (Parliament), setting up charities - and in government she has that experience".
The 53-year-old is keen to encourage the Thatcher comparison.
In reference to Brexit, 44 per cent of those surveyed said they thought May would get the best deal from the European Union, compared to a quarter who sided with Leadsom and 16 per cent with Michael Gove.
An obscure politician until she shot to prominence as a pro-Brexit debater, Leadsom is trying to shed the image that she is a novice.
But it was - in truth - the ideal symbol of the fact that she and her backers see themselves as part of a peasants' revolt against the Tory establishment and David Cameron's project of Tory modernisation.
He also celebrated the fact that the Tories will have been responsible for both of Britain's female prime ministers, calling the center-right party "the most progressive party in Britain".
But she stopped short of backing Mr Crabb, who stepped down from the race after the first round of the ballot, saying she wanted to wait to see who MPs chose before revealing who she would support. She denies the claims.