Brexit Secretary Steve Barclay refused to rule out running for the leadership.
He was asked by Sky’s Sophy Ridge four times if he would be standing – and four times he failed to rule it out.
Mr Barclay is joining what is already a crowded field.
Mrs May effectively fired the starting gun on the race to succeed her last week when she agreed to meet Tory chiefs in the first week of June to finalise a timetable for her departure.
But he added: “I would say to my colleagues – focus on the day job, concentrate on what is before us, ensure we deliver on what we promised to the public, ensure that we leave.”
Asked if Mr Johnson would be a good PM, Mr Barclay replied: “All of my colleagues have talents.
“It’ll be for others to weigh up which of those takes primacy.”
The number of Tory MPs suggesting they will run has led to comparisons with the Grand National with some commentators joking that it is easier to remember who is not standing.
Former Foreign Secretary Boris Johnson is 11/4 to claim the Conservative crown, three years after abandoning his bid in the wake of the Brexit referendum.
Other declared candidates include former Brexit Secretary Dominic Raab, newly-appointed International Development Secretary Rory Stewart and former Work and Pensions Secretary Esther McVey.
But there are also a whole host of others who are yet to declare.
Chief Secretary to the Treasury Liz Truss, Defence Secretary Penny Mordaunt, Home Secretary Sajid Javid, Commons Leader Andrea Leadsom, Work and Pensions Secretary Amber Rudd and Foreign Secretary Jeremy Hunt are expected to join the contest.
Ruling himself out of the race, Defence Minister Tobias Ellwood told Sky News: “I don’t have the rank and I don’t have the experience.”
The former Army Officer warned colleagues vying to replace Theresa May : “This isn’t a TV reality contest.”
And he urged potential leaders: “Let’s have some loyalty, let’s have some commitment, let’s have some discipline in focusing on the one thing that we need to get across the line, which is Brexit.
“The idea that we jump ahead and start toying around with who should be the leader… I do urge my colleagues to think why are they putting their name forward?
“Is it to raise their own profile? And some colleagues as well are changing their views in order to woo our membership to raise their profile as well.
“This is getting things in the wrong order.
“Our focus must be to get Brexit across the line.”