The vote, which was delayed at the last minute in December, is currently scheduled for the week beginning Jan 14.
May will begin a charm offensive next week by inviting every Tory MP to Downing Street for drinks parties on Tuesday and Thursday, in the hope that she can win over those who doubt her Brexit deal.
Whips have been hard at work over the Christmas break contacting Tory rebels individually to discuss their specific concerns about the deal, but senior Brexiteers said nothing had changed since Parliament went into recess last month.
Yesterday May spoke to Angela Merkel, the German Chancellor, who is seen as the key to unlocking a more favourable deal that could win the support of Parliament.
May wants legally-binding assurances from the EU that the backstop arrangement to prevent a hard border in Ireland in the event of no future trade deal being agreed would only be temporary.
She also wants Britain to have a unilateral exit mechanism from that backstop.
May is expected to return to Brussels before the vote takes place, but Davis speculates that the vote could be put off for a second time if May looks likely to lose it.
He says: "The Withdrawal Agreement does not respect the referendum result. That is why the meaningful vote had to be delayed and one wonders if even the January vote will go ahead.
"Attempts to frighten MPs into supporting it are unlikely to work, because voting down this substandard deal will not result in no Brexit."
Urging May to take her time to get a better deal, he adds: "We know that the EU is worried about the loss of the £39 billion 'divorce' payment if there is no deal... so this is the moment to be hard-nosed about these issues. The more we prepare to leave the EU without a deal, the more likely a good deal becomes."
Nigel Dodds, Westminster leader of the DUP, on whose votes May relies for her majority, was in Downing Street for a Brexit-focused meeting in the Chief Whip's Office.
A DUP spokesman made it clear the party still intends to vote down the deal unless it is changed, saying: "The DUP has been consistent in these negotiations."
Speaking in Singapore, Hunt appeared to suggest that no-deal was not a serious prospect, as he warned against the "disruption" that it would lead to.
He said: "We have to remember that a no-deal Brexit would cause disruption that could last some time. Even if you are someone who believes that Britain will flourish and prosper whatever that disruption might be, that is not something that any government should willingly wish on its people.
"A second referendum would be also incredibly damaging in a different way... the social consequences of not going ahead and leaving the EU on the 29th of March as we have been instructed to do would be devastating."
Hunt, who supported Remain in the 2016 referendum but says he would now vote Leave if the referendum happened again, said May was still talking to EU leaders to improve the deal currently on the table.
In particular, she wanted to make sure that the UK would not be "trapped" indefinitely in a customs-union with the EU, he said.
This would mean fresh concessions over the Northern Ireland backstop.
He added: "I think she [May] will find a way to get this deal through Parliament and think that is what the British people would want."
Speaking on BBC Radio 4, Hunt said MPs could get "absolutely everything we want" from May's Brexit deal.
He said: "We have a clear opportunity to leave the EU on March 29. It has the vast majority of things that people wanted, not absolutely everything.
"The question is, can we turn this into something that gives us absolutely everything we wanted, and I believe we can," he declared.
"There will be some tough negotiations to follow in the years ahead but I think getting this clearer language on the backstop will help to get it through Parliament."
