Meanwhile, it can be revealed that the October march organised by the People's Vote was attended by just a third of the number that its organisers claimed, according to an internal estimate by the Greater London Authority.
Senior EU figures are said to be waiting until the last moment before the deadline for reaching a Brexit deal to offer any additional measures, so MPs are left with no time to demand more.
May's advisers are understood to be considering an amendment making approval of the deal subject to the Government obtaining assurances that the Irish backstop, the fallback plan intended to prevent a hard border, will be temporary.
Nikki da Costa, who was Downing Street's director of legislative affairs until November, said: "Getting conditional approval isn't enough for the Government to go ahead but it may be enough to show the EU there is a majority if they can move a little further."
She added that such an amendment could allow the Government to "return for a third go at the meaningful vote, with a proven recipe for a majority."
The vote was initially planned for last month but was postponed in the face of an expected defeat of up to 200 MPs. It was next due to take place on Jan 15 or 16, after debate on the deal resumed in the Commons this week.
Without a way for the UK to extract itself from the arrangement, the amendment would be unlikely to win over the DUP's 10 MPs on whose votes Mrs May relies.
Another possible amendment would give Parliament the right to serve notice to exit the backstop after a year if the sides failed to agree a trade deal to resolve the border issue.
-- The Sunday Telegraph