Mr Gauke, who has served in Theresa May‘s cabinet since she took office in June 2016, said he would not be able to serve under the former foreign secretary if he pursues a no-deal Brexit.
As the Tory leadership contest enters its final 48 hours, the prominent no-deal critic told the Sunday Times that crashing out of the European Union would lead to national “humiliation”.
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And Mr Gauke said: “Given that I’ve been in the Cabinet since Theresa May came to power, I think the appropriate thing is for me to resign to her.”
The paper also reported that up to six Tory MPs are considering defecting to the Liberal Democrats should Mr Johnson beat his rival Jeremy Hunt and enter Number 10 – leaving him with no majority in the Commons.
The ballot for the Tory leadership race will close on Monday, with the result announced the following day.
Mr Johnson, who is widely predicted to win the contest, has reportedly been secretly wooed by European politicians and officials in a bid to thrash out a new Brexit plan to avoid a no-deal exit, the Sunday Times reported.
It said figures from Ireland, France, Germany, Belgium and the Netherlands had established contact with the frontrunner’s team and signalled their intention to do a deal.
Amber Rudd, the work and pensions secretary and former Remainer, urged her Tory colleagues to back whoever wins the contest.
Writing in the Sunday Telegraph, she said: “Just as the Conservative Party came together after that EU referendum, it must now come together again no matter who wins.
“I know many of my colleagues have strong feelings but a new prime minister will get a new hearing in the European Union and I hope everyone can unite behind that.
“If we don’t, the alternative could be Corbyn for Christmas.”
Her comments came after Philip Hammond, the chancellor, fired a warning shot to Mr Johnson, saying he would do “everything in my power” to block a no-deal Brexit and left open the possibility of voting to bring down a Conservative government in such an event.