Separately it emerged that Gina Miller, who brought the lawsuit that forced Mrs May to get parliamentary approval for her Brexit plans, is considering another legal challenge if Parliament is not given a full vote on the final deal.
British Ministers are expected to try to overturn an embarrassing defeat in the House of Lords on the Government's Brexit Bill when the legislation returns to the House of Commons.
But a spokesman for the Prime Minister said she remained "committed" to meeting the deadline to trigger Article 50 by the end of March.
Labour and Liberal Democrat opposition parties are joining forces with rebels among May's Conservatives to support an amendment to protect Europeans residing in the UK.
"To continue to use people as bargaining chips in this way is not only shameful but could have a dire impact on the UK's economy and essential services", said Baroness Angela Smith of Basildon.
Mr Davis also claimed MPs "may have a different view" to the House of Lords as well as repeating the Government's insistence the vote would cause "no change" to the Prime Minister's promise to trigger Article 50 by the end of March.
While Britain hasn't yet triggered Article 50 of the Lisbon Treaty, which would begin its exit from the EU, the non-British European nationals who call the United Kingdom home are already anxious about their futures - particularly highly skilled workers in sectors that often rely on EU funding and collaboration with institutions on the continent.
If the Commons then backs the amendment it would weaken May's threat to Brussels that she could walk away if it refuses to play ball since "no deal is better than a bad deal".
While Mrs May has promised a vote on any agreement with the European Union, opposition parties as well as some pro-Remain Conservatives, fear that if there is no deal Britain would simply crash out of the European Union with Parliament unable to intervene.
Given that there has been a referendum, and given that the bill was passed without amendments by the elected Commons, any attempts at further stalling by the Lords would raise questions about their democratic legitimacy.
And for the first time ever, Mr David Davies, the British minister responsible for negotiating the separation from the European Union, has publicly warned his Cabinet colleagues "to prepare not just for a negotiated settlement" but also for the possibility that "no mutually satisfactory agreement can be reached".
May had hoped to agree a deal on EU nationals late previous year, but European leaders insisted there could be no talks on any subject before the government triggers Article 50.