Mr Baker, a former Brexit minister, is an influential figure in the European Research Group (ERG) of anti-EU Tories. On Thursday he was among backbenchers who demanded a vote of no confidence in Ms May, and called on others to do the same.
Eurosceptics are believed to be close to the threshold of 48 signatures needed to secure a no confidence vote in the prime minister.
With two cabinet ministers already having resigned in protest at the proposed deal, Mr Baker said others must follow suit if they are to realise their leadership ambitions.
He told Sky News: "For those people who are in Cabinet today, it is perfectly noble and reasonable of them to prioritise the stability of the government, to prioritise - particularly for Michael [Gove] - no deal prep, because Defra is one of the more affected departments.
"That is perfectly reasonable and noble. But it does mean, I think, that they will not be one of the plausible candidate should a contest come up."
He said Eurosceptic MPs would not put forward ERG leader Jacob Rees-Mogg as their candidate because he has "no experience in government". He also ruled himself out of running.
Instead, he said he wanted to see a candidate who is "charismatic and determined, who believes in leaving the European Union and who can carry this country through it".
Both Mr Gove and Ms Mordaunt were reported to be considering their positions following the resignations of Brexiteer colleagues Dominic Raab, the Brexit secretary, and Esther McVey, the work and pensions secretary.
Both, however, have since decided to stay. Reports suggest they will join with fellow Eurosceptics Liam Fox, Chris Grayling and Andrea Leadsom in trying to convince Ms May to shift her stance on Brexit.
Mr Baker is understood to be co-ordinating letters calling for a vote of no confidence in the prime minister.
He said he did not know exactly how many had been sent to Sir Graham Brady, chair of the 1922 committee of Tory MPs, but added: "My count is a little over 48 with perhaps another dozen probables."
The current situation was "extremely grave", he said, adding: "It gives me no pleasure to be doing what we're doing".
In a scathing attack on Ms May, Mr Baker said Tory MPs must vote to oust the prime minister if they want the party to win the next general election.
He said: "The thing that people will need to remember is that we went into [the 2017] general election with a poll lead so great that the serious concern of journalists and commentators was that we would be going into a one-party state.
"That was all thrown away and every member of Parliament has had to live this awful hung parliament, where if half a dozen MPs decide they don't like any policy, you can't get it through, and the government has to fold again and again and again.
He added: "This is not the way our country needs to be governed, and it's governed this way because we had a disastrous general election.
"Every [Conservative] member of Parliament will face a secret ballot if the vote comes, in which they will have to ask themselves, 'do I want to repeat that?' and the answer for most of them is going to be no."