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Theresa May will resign on June 7.
MPs in Kent have already pledged their support to senior Tory MPs ahead of a leadership battle.
She made an announcement outside Downing Street this morning after talks with senior Tory MP Sir Graham Brady, who heads up the group representing Tory backbenchers.
In an emotional speech, she said she would leave “with no ill will, but with enormous and enduring gratitude to have had the opportunity to serve the country I love”.
But she admitted: “It is and will always remain a matter of deep regret to me that I have not been able to deliver Brexit."
While she had done "everything I can" to gain support for her Brexit deal, it was now in the "best interests of the country for a new prime minister to lead that effort".
One Kent MP has already hinted who he thinks should be the next leader of the party - ahead of the expected resignation of Theresa May.
Tonbridge and Malling MP Tom Tugendhat says he is minded to give his backing to Michael Gove, the environment secretary of state and former education secretary of state.
Mr Tugendhat revealed his preferred choice after he publicly called on Theresa May to quit. In an interview with the Financial Times, he said: “Leadership matters and it has been absent for too long.”
He had backed the government’s Withdrawal Bill on the three occasions it came before the Commons.
He said he favoured Mr Gove because he had wide-ranging experience in key government roles: “I’m minded to support somebody who has demonstrated huge capability in various departments. Who has demonstrated the intelligence and ingenuity to overcome many challenges.
"That’s why I’m more minded towards Michael Gove. But let’s see who starts this race, nobody has officially put their name in the ring yet.”
“I’m minded to support somebody who has demonstrated huge capability in various departments..." Michael Gove
He was cool about the leadership bid by Boris Johnson, telling LBC he was looking for somebody who has demonstrated an "impressive record of achievement," pointing out that Michael Gove did "all the debating" on the leave campaign.
The MP has been tipped as a potential future leader but his backing for Mr Gove is an indication that he will not be putting his hat in the ring - at least this time.
Mr Gove was criticised for abruptly ending his support for Boris Johnson during the 2016 leadership contest and putting himself forward for the job.
Ashford MP Damian Green said his overwhelming feeling was one of sadness at the resignation of Theresa May: “It may be that the European issue is one that arouses so much passion and visceral views on both sides that any amount of schmoozing and on squeezing would not have made any difference.”
He said that the key turning point was when David Davis resigned as brexit secretary. But he rejected the suggestion that her personality meant she was not suited to being Prime Minister.
Speaking on the BBC Today show he said:“Perhaps inevitably people are concentrating on the downside but we have to remember the extraordinary sense of public service in an era when political discourse has become so poisonous, she was always courteous and polite.
"The biggest tragedy is she never had the time or place to implement her domestic agenda and tackle the burning injustices she wanted to address."
MP for Maidstone and the Weald Helen has resigned and backed Dominic Raab.
Theresa May's appeal for a successor who would be willing to compromise on Brexit did not go down well with one Kent MP.
South Thanet MP Craig Mackinlay said: “We need a strong Brexiteer as leader, there is no room for a compromise here, we have to have someone who can deliver Brexit. We have seen that compromise has not worked." He admitted the party was not "in a good place" but would not be drawn on who should be her successor. But he was not convinced Michael Gove was the right person.
There was support for the leader from Sir Roger Gale, North Thanet MP. He tweeted:
Dartford MP Gareth Johnson said he would be backing former Brexit secretary Dominic Raab, who is yet to confirm that he will stand.
"It is the case she tried her best but if you say a hundred times that you are going to leave by March 29 with or without a deal and that doesn't happen there is a consequence that comes with that and we saw that happen today."
He said Mr Raab had the ability to unite leavers and remainers and had the intelligence to do the job. "It is a difficult time for us and we need someone with those abilities."