Jenkyns said Steve Baker, a junior minister in the Brexit department, had also quit.
BBC political editor Laura Kuenssberg said she understands Mr Davis was "furious" after a meeting at Number 10 earlier on Sunday and "concluded he could not stay in post".
Davis's late-night resignation undermined May's fragile government.
And he repeated his words from Chequers that the "common rule book" proposal "hands controls of our economy to the European Union and is certainly not returning control of our laws in any real sense".
Following a long weekend which started with extended talks over the country's future relationship with the European Union at the Prime Minister's Chequers country pad, an agreed strategy met the backing of her Cabinet.
Davis and Baker, both longstanding euroskeptics, decided they couldn't support the policy, a person familiar with the matter said.
The long-serving Conservative MP was among the new intake when Tony Blair's new Labour swept to power in 1997.
The pound weakened slightly on the news. "Will those red lines still there survive negotiation with the European Union?"
Mr Davis's resignation will anger cabinet colleagues from both wings of the party.
Former MP for Peterborough Stewart Jackson has left his role as a special adviser in the Government following the surprise resignation of the Brexit Secretary.
In the Commons on Monday Mrs May is expected to tell MPs that the strategy agreed on by the cabinet at Chequers on Friday is the "right Brexit" for Britain. Some lawmakers have already expressed their misgivings. Tory sources suggested a leadership challenge was on the cards, perhaps this week. Although UK politics is volatile, the pro-Brexit camp in Parliament would likely struggle to get the numbers together to win if May chose to stay and fight.
- May leads a minority government which has only a slim majority in parliament thanks to a deal with a smaller party.
Britain's Prime Minister Theresa May and her cabinet discuss the government's Brexit plans at Chequers, the Prime Minister's official country residence, near Aylesbury, Britain. If she can't reassert her authority, the move could trigger a chain of events that result in a general election.
She said the government's Brexit plan was "far from perfect" but marked "grown-up steps".
He told BBC 5 Live: "These proposals will have to come to the House of Commons in legislation and the question is 'will they command support from Conservative MPs?'".
"This chaos is good for no one. And one of the things about this compromise is that it unites the Cabinet".
An analysis of the Chequers deal circulating within the pro-Brexit European Research Group of MPs was damning about the plan, saying it would lead to "a worst-of-all-worlds "black hole" Brexit where the United Kingdom is stuck permanently as a vassal state in the EU's legal and regulatory tar pit".
"If the proposals are as they now appear, I will vote against them and others may well do the same", he said in an article for Monday's Daily Telegraph newspaper.