The Conservative party leadership contest was triggered by Cameron's decision to resign following the June 23 public vote to leave the European Union (EU).
With the second round of voting among Conservative lawmakers complete, May and Leadsom — baring any last-minute withdrawals from the race — will face a vote by the entire party membership.
A final winner is expected to be announced by September 9 and Cameron remains leader and prime minister until then.
The next leader will have to take on the tricky position of triggering Article 50 of the Lisbon Treaty, which sets in motion the withdrawal process from the EU and will have to negotiate Britain's tricky exit from the bloc.
Home Secretary May is the favorite to succeed Cameron. Unlike Leadsom, she supported Britain staying in the EU, prompting some hopes that she might find a more sympathetic reception in Brussels during withdrawal talks and that she might take a softer stance to what Britain's relationship to the EU will look like post-Brexit.
Still keen to disabuse anyone (least of all pro-Brexit Tory party members) of the notion that she will renege on the result of the referendum, May has reiterated so far that "Brexit means Brexit" and she will strike a hard bargain with EU leaders over the terms of withdrawal. For their part, EU heads of state want Britain to stop dithering and to trigger Article 50 in order to curtail a period of economic and political uncertainty brought about by the Brexit vote.
The other main candidate, Leadsom is a keen "Brexiteer," saying on Thursday that she will make Britain "the greatest nation on earth" – something looking like a harder task as the pound struggles to regain its footing and as businesses go into "wait and see" mode in terms of investment and hiring.
Gove, who was knocked out in the second round, was mired in controversy – first for betraying fellow Brexit campaigner and former London Mayor Boris Johnson by withdrawing his support for Johnson's leadership hopes and deciding to run himself and secondly, after it emerged that his campaign manager had reportedly texted Theresa May's supporters to back Gove instead.
In a first round of voting on Tuesday, Liam Fox, the former defense secretary, was eliminated from the contest with the least amount of votes while Work and Pensions Secretary Stephen Crabb pulled out of the contest and endorsed May.
—With contribution from CNBC's Katy Barnato. Follow CNBC International on Twitter and Facebook.