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And he adds his call for a "peoples' vote"...

Laura Kuenssberg
Â鶹ԼÅÄ political editor

There's been no official comment yet from Mr Davis' fellow prominent Brexiteer Boris Johnson, our political editor notes.

But the foreign secretary is due to speak later this afternoon at a "long-organised" press conference at the Western Balkans Summit in London.

Perhaps we will get comment from him then?

Laura Kuenssberg
Â鶹ԼÅÄ political editor

The Â鶹ԼÅÄ's political editor, Laura Kuenssberg, said a Brexiteer source told her after the Chequers summit their anger "did not have a home".

"Well, [Davis'] resignation provides them with a home," she added.

And whilst Mr Davis denied he was trying to weaken the PM, Laura said he had just given a "very serious criticism of Theresa May's main policy" on live radio.

She said: "The man who has been intimately involved in the negotiations just told us his view is this type of Brexit does not work."

She also said there were backers of Davis "willing to put in letters today for a vote of no confidence in the prime minister" - but it was not yet clear if there were enough to force one.

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When asked if he would make a leadership challenge, Mr Davis says: "Leadership? No I would not.

"I would not throw my hat in the ring.

"If I wanted to bring down Theresa May the time to do it would not be now, it would have been after the election."

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Davis Davis denies his decision will weaken the prime minister.

In fact, he says: "I like her, I think she is a good prime minister."

He claimed his leaving is just "one day", adding: "She has got to have a Brexit secretary that will deliver on her strategy."

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Mr Davis says he hopes his resignation will put "pressure on government not to make any other concessions".

"I will keep arguing there are better ways to do it than this," he says.

David Davis said it was often misunderstood that Theresa May was the "primary negotiator" with the EU, rather than him.

"People often don't realise this, that one of the first things she said to me is she would do the primary negotiations," he told Today.

When asked if he had been "cut out" of the process, he denied it.

Mr Davis said he proposed one approach, but Mrs May chose another "more conciliatory" one.

He added: "My fear is they will take what we offered already and want some more.

"It seems to me we are giving too much away too easily and that is a dangerous strategy".

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Mr Davis said he did not expect others to follow his lead by resigning.

He said it was "different" for him compared to other cabinet Brexiteers, such as Boris Johnson, because he was the one that had to lead on the policy.

He said: "The simple truth is that there are many things you don't agree with in cabinet... that's normal.

"But if they are central to you, your own policy, your own department, your own delivery, then it becomes more sharped edge... more central to you."

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He continues that it was also known he had "concerns about" Theresa's May's Brexit plan.

He says he told Mrs May: "I hope you’re right, I hope I’m wrong."

But he says the "best person" for the Brexit Secretary job would be someone who believes in Theresa May's plan.

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David Davis said he first discussed the new Brexit policy on Monday last week and was "not at all happy with it".

He then read it on Wednesday and raised issues with the PM.

He said there was a "forthright debate" on Friday, but Mrs May "won the argument by two or three to one".

However, he took two days to make his decision as didn't want to rush out in a cabinet meeting and "do a Michael Heseltine" he says.

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David Davis says his wife was among those he spoke to before this decision to quit.

"I took two days to think it through - a very important decision," he tells John Humphrys.

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You can listen live here.

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Health Secretary Jeremy Hunt told the Today programme he does not believe there will be a vote over Theresa May's future.

He said there was a clear mechanism for forcing a vote within the party but "I don't believe that will happen".

"I hope that doesn't happen. It's now or never. If we are going to deliver the clean Brexit that people want, we need to get behind Theresa May," he said.

He was asked whether he wanted to apologise - on behalf of the Tory party - for "the mess". Mr Hunt said that was a "ridiculous thing to ask".

And in reference to that infamous Â鶹ԼÅÄ vox pop ahead of the last general election, the Sun's deputy political editor tweets:

Image source, PA

Health Secretary Jeremy Hunt praised his former cabinet colleague.

He said when Brexit proves to be a success, "history will judge him to be one of the key architects", adding that he "piloted the EU Withdrawal Bill through the House of Commons".

"All of us feel a sense of great loss," said Mr Hunt, who campaigned for Remain.

He called Mr Davis an "outstanding colleague" and a "thoughtful man with real integrity".