But a key amendment empowering MPs to stop the United Kingdom from leaving with no deal, or to make Theresa May return to talks, has been rejected.
But he said: "It is now essential that government makes progress on our long-term customs and other border arrangements".
A source close to Mr Davis said there had been "a back and forth" on the wording of the paper, which "now expresses, in much more detail, the time-limited nature of our proposal".
The government decided not to change an amendment that would force it to try to negotiate a customs union with the EU that is shaping up to be the most contentious vote in parliament next week, an official with the opposition Labour Party said.
These figures produced a total cost for UK-EU trade in goods of about £17bn-£20bn a year, he said.
"If we move to WTO (World Trade Organization) rules, that would definitely require customs declarations so it would be similar in terms of costs", Thompson said when asked about the cost of a no-deal Brexit.
At a press conference in Brussels, Barnier also warned that May's proposal for a strict time limit to apply to any backstop plan would not be acceptable.More news: Argentina call up Perez to replace Lanzini after West Ham ace's injury
Following his around half hour-long meeting with Mrs May, Number 10 revealed the prime minister expected Mr Davis to remain in his position.
The comments followed speculation at Westminster that Mr Davis had considered quitting over a dispute regarding backstop proposals on how to deal with the issue of the Northern Ireland border if a preferred withdrawal trade deal is not sorted out with the EU.
"Instead, with the threat of a cabinet resignation, Theresa May has signed up to a flawed proposal which is inconsistent with her earlier commitments".
Should Britain's proposal be unworkable, European Union nations insist that a "backstop" remain in place to keep Northern Ireland within Europe's customs arrangements.
The future of the border on the island has dominated the Brexit debate in recent weeks with ministers on Thursday finding a compromise on the thorny issue of a backstop Brexit plan for the Irish border.
On Friday, a spokeswoman for Prime Minister Theresa May said simply: "There isn't going to be a second referendum".
PM May also faces a series of votes in the House of Commons on Tuesday 12, June on whether to approve 15 "Brexit wrecking" amendments inserted into the EU Withdrawal Bill by the House of Lords. But pro-EU legislators from both the Conservative and Labour parties say they will push for the U.K.to remain inside the single market by joining the European Economic Area, a grouping of the EU plus Norway, Iceland and Lichtenstein.