He said “serious questions and concerns” were raised by Mr Farage’s claim that he was offered a peerage three days before standing down candidates in 317 Tory-held seats.
Mr Davey also claimed that the Lib Dems were the “main challengers to the Tories” in many of the affected constituencies.
Outlining three key questions, he first asked Mr Johnson to confirm “whether Nigel Farage or any Brexit Party candidate or official was offered a peerage or an official appointment for allying with the Conservatives during this general election.”
He then urged the prime minister to confirm whether the Brexit Party leader – a friend of Donald Trump – was offered the role of British ambassador to the US.
“Such a move would be of grave concern given the subsequent increased threat of a post-Brexit UK-US trade deal that would undermine our NHS, farmers and other businesses,” Mr Davey said.
His third and final question to Mr Johnson requested confirmation on whether he or his party had taken legal advice on the Honours (Prevention of Abuss) Act of 1925, which banned the sale of peerages.
The Liberal Democrats have also called on the Conservatives to launch an independent internal enquiry into allegations officials offered jobs and peerages to Brexit Party candidates in return for standing aside.
Sal Brinton, the party’s president and a member of the House of Lords, said: “The severity of these claims has implications not just for the integrity of the General Election, but also for the public’s trust in our democracy.”
The Metropolitan Police has confirmed they are assessing “two allegations of electoral fraud and malpractice in relation to the general election”.
Mr Johnson has admitted there were “conversations” between senior Tories and the Brexit Party but denied that peerages were offered, saying it was “just not the way we operate”.