The Brexit Secretary said negotiations are expected to go on up to the last minute of the final day before Britain quits the EU.
Mr Davis said Brussels' negotiating track record meant talks would be pushed to the wire, but it would be "very exciting for everybody watching".
It would mean MPs would only get to vote on any deal that was thrashed out after Britain had left the bloc, he told the Commons Exiting the EU Committee.
His comments are likely to anger MPs who believe parliament should get to sign off on any deal before the UK leaves the EU. An amendment to the European Union (Withdrawal) bill already supported by Conservative backbenchers calls for a Commons vote before March 29, 2019, setting up a new battle with the government when the legislation progresses.
Mr Davis said: "It's no secret that the way the union makes its decision tends to be at the 59th minute of the 11th hour of the 11th day and so on, and that is precisely what I would expect to happen."
He added: "If there is a time limit on a negotiation the union stops the clock, it assumes that it's still at 11:59 until it is concluded, sometimes over the course of 24, 36, 72 hours thereafter and that's what I imagine it will be.
"And it will be a lot of pressure, very high stress, very exciting for everybody watching."
Theresa appeared to confirm Mr Davis' timetable when she was asked about his comments at Prime Minister's Questions.
Stephen Kinnock, the Labour MP, asked the Prime Minister to "please explain how it is possible to have a meaningful vote on something that has already taken place?"
Mrs May said she was "confident" "The timetable under the Lisbon Treaty does give until March 2019 for the negotiations to take place.
"I am confident, because it is in the interests of both sides - and it's not just this parliament that wants to have a vote but there will be ratification by other parliaments - that we will be able to achieve that agreement in time for this parliament to have the vote that we've committed to."
The Prime Minister's official spokesman later added that the target for agreeing the UK's Brexit deal was October 2018, and that MPs would be given a vote, but did not say whether it would be held before or after the UK leaves the EU.