Sir Vince Cable has hit back at Boris Johnson after a source close to the foreign secretary accused the Liberal Democrat leader of lying.

The row started when Cable, a former business secretary, reportedly claimed Johnson was planning to quit Theresa May's government as Brexit in-fighting continues in the cabinet. Johnson's office vehemently denied the claims, accusing Cable, 74, of "peddling lies".

"Boris and [Chancellor] Philip Hammond are working closely to take the UK out of the EU and are not going to be diverted from that important task," the source said.

But Cable, who succeeded Tim Farron on 20 July, refused to apologise and used the opportunity to raise doubts about the government.

"If the Foreign Secretary is working so closely with the Chancellor, can he confirm that he supports the Chancellor's call for a three year transitional deal with continued free movement of labour? So far he has been uncharacteristically silent on the subject," he said in a statement released to the Press Assocation.

"It is a simple question: does he support the position of Philip Hammond or [international Trade Secretary] Liam Fox? Because he can't support both. And if Philip Hammond secures a three year transitional deal, can Boris Johnson confirm he will stay in the government and support the policy?

"I have a policy: it is to stay in the single market and customs union to protect British jobs and living standards."

The dispute between Cable and Johnson comes amid alleged rifts in the cabinet over a potential Brexit transitional deal, with Hammond saying the UK could finally split from the EU in 2022 after David Davis said Britain would split from the EU's single-market and customs union in 2019.

Hammond, meanwhile, has told French newspaper Le Monde that the UK will not slash tax or regulations after its split from the EU.

"This government has broken down into farce," said Peter Dowd, Labour's shadow chief secretary to the Treasury. "The Chancellor is not only disagreeing with Cabinet colleagues over Brexit, he is now in open dispute with himself given it is only his own comments on the matter in January which he is pretending to contradict.

"The truth is that the British people will not believe the fake U-turn of a Tory Chancellor in a French newspaper, while he is still going ahead with billions of pounds in corporation tax giveaways in this parliament, and refuses to rule out further cuts.

"Only a Labour government will end the Tory tax giveaways to the super rich designed to turn our country into a tax haven on the shores of Europe, and bring about a Brexit for the many, not the few."