May's Conservative government is split between ministers who favor a clean-break "hard Brexit" - leaving Britain freer to strike new trade deals around the world - and those who want to keep closely aligned to the EU, Britain's biggest trading partner. "The temporary backstop is not in line with what we want or what Ireland and Northern Ireland want and need".
The debate will test May's ability to broker a compromise with those in her Conservative Party who, like many members of the upper house, want to keep a relatively close relationship with the European Union after Brexit.
At the heart of the problem is ensuring there is no hard border between the British province of Northern Ireland and the Irish Republic, which will remain in the EU.
Barnier will meet David Davis, the UK's chief Brexit negotiator, on Monday and said he will raise these points at the meeting.
"The PM has been clear that we will preserve the integrity of the U.K.'s internal market and there will not be a customs border down the Irish Sea - any backstop option will satisfy this", he said.
London says this is unacceptable, offering instead a "temporary customs arrangement" for the whole of Britain that would "maintain the status quo for traders in respect of customs processes".
But he said: "It is now essential that government makes progress on our long-term customs and other border arrangements".
The development came as Mrs May braced herself for possible defeat in the Commons next week on an amendment which would require her to try to negotiate a permanent customs union with the remaining EU.
Brexit talks have been stalled in Brussels over the question of how to avoid a hard border at the Irish frontier.
A spokesman for the United Kingdom government said: "Our solution to the border issue is the deep and special partnership with the European Union that we are confident of negotiating".
The source close to the government and some local media said Davies was considering whether to resign.
Rather than setting a firm deadline for the end of the backstop, the document states: "The UK expects the future arrangement to be in place by the end of December 2021 at the latest".
Claiming there is "clearly a great deal of work" remaining to be done in Brexit negotiations, Mr Coveney said: "Our strong preference remains an overall EU-UK future relationship which would resolve all issues".
An anti-Brexit pressure group has launched its formal campaign for a second European Union referendum, publishing its roadmap to delivering a "people's vote" in 2019.
Barnier said the UK's proposal for keeping open its border with Ireland after it leaves the bloc in 2019 also appeared to be geared toward extending it to the whole of Britain - something Barnier said could not be done.