Sources close to Lee said his resignation had been a "warning shot" to the Government and that more departures would come if May had not made further concessions on her plans for a so-called "meaningful vote" and customs arrangements by the time two more Brexit Bills are debated next month.
Yesterday, on a day of drama and brinkmanship in Parliament, May avoided defeat after a concession to Tory rebels over the meaningful vote on the final Brexit deal.
Whips and ministers shuttled back and forth for hours during a debate over the EU Withdrawal Bill in a frantic attempt to reach a compromise with Remainers.
Dominic Grieve, the former attorney general who led the rebellion, claimed he had killed off the prospect of a "no deal" Brexit and suggested it was Lee's resignation that had triggered the climb down, but the Government insisted no such concession had been or would be made.
Details of Best For Britain's attempts to thwart Brexit were disclosed by The Daily Telegraph in February, when it emerged that the group had plans to "pressure" MPs in 100 Leave-supporting constituencies.
The group claimed it had been in contact with Lee before his resignation.
Lee called for a second referendum in his resignation statement, and Best For Britain claimed it had been in contact with other ministers it believed to be sympathetic to a second vote, including Alistair Burt, the Foreign Office minister, Richard Harrington, the business minister, Shailesh Vara, the Northern Ireland minister and Rachel Maclean, a parliamentary private secretary to the Home Secretary.
Harrington and Maclean denied having meetings or contact with Best For Britain.
Eloise Todd, the chief executive of Best for Britain, said: "We have been speaking both to Government ministers and opposition MPs to make our case and lobby them on Brexit."
Vara said: "I'm not thinking of resigning. I'm absolutely a solid Brexiteer and if there was a referendum now I would campaign to leave. I don't agree with their campaign, the people of Britain made their voice absolutely clear."
Burt said that he had "absolutely no intention" of resigning over Brexit.
A spokesman for MacLean said she had never spoken to Best for Britain or attended an event held by them, describing suggestions she could resign over Brexit as "utter rubbish".
Harrington issued a "categoric" denial that he had ever met anyone from Best for Britain and said he would not resign.
Lee was unavailable for comment.