The MP for for Worcester is one of two ministers left in post at the Department for Exiting the European Union (DExEU) after the resignations of Mr Davis, his deputy Steve Baker and the reported departure of Suella Braverman. "And how has our Government secured Gibraltar's corner specifically 24 months on?"
May has spent the past few days fighting for her political life as first Brexit Secretary David Davis and then Foreign Secretary Boris Johnson quit, saying May's plans for future relations with the European Union did not live up to their idea of Brexit.
Davis and Johnson initially backed the plan, before deciding they could not support it.
Just 30% of Britons think Theresa May is the best Conservative to lead Brexit negotiations, with 44% saying someone else within the party would be better suited, and 26% answered that they don't know.
Fianna Fail's Brexit spokesperson Lisa Chambers said the double resignation was of "grave concern" but urged the British Government to continue with its plan to publish the white paper this week.
Asked Tuesday whether May should be replaced as prime minister, Trump said it was "up to the people, not up to me".
And senior backbencher Bernard Jenkin said there had been a "massive haemorrhage of trust" in Mrs May.
But MPs from the rival Labour Party attended a briefing on its specifics-a sure sign that May thinks they may be needed to get it passed through parliament.
The ERG's chairman, Jacob Rees-Mogg, has said he has not sent a letter to the 1922 Committee, and expects Mrs May to remain in office at least until Brexit Day in March 2019.
A handful of lawmakers who want a clean break from the European Union said privately they had submitted letters calling for a vote of confidence in May, though hadn't met the threshold of 48 needed to trigger such a ballot.
Sir Graham refused to say whether he had received any such letters.
Asked whether May would contest a no-confidence vote, a Downing Street source said simply: "Yes".
Johnson had been due to host a summit about the western Balkans on Monday afternoon but was instead holed up in his official residence with close advisers, considering his position.
Interrupted only by a near-death electoral experience when Conservatives lost their majority in parliament in June 2017, May's tenure has been a never-ending party feud marked by frequent and intemperate public statements from several cabinet ministers, especially her now-former foreign secretary, Boris Johnson. "The Prime Minister thanks Boris for his work".
Dominic Raab was appointed Brexit secretary by Mrs May after Davis' resignation.
"Boris Johnson is a friend of mine, he's been very, very nice to me", Trump told reporters before heading to Brussels for a summit of North Atlantic Treaty Organisation allies.
Some euroskeptic lawmakers dream of replacing May with a staunch Brexiteer such as Johnson, a populist, polarizing figure who has never made a secret of his ambition to be prime minister.
Mrs May surprised many by appointing him to the Foreign Office, a position he has frequently used to forge a distinctive position on Brexit, including by setting out his own "red lines" just days before the PM's crucial speech to the Conservative conference previous year.
In her response to Johnson, May wrote that she was "sorry and a little surprised" by his move after his apparent support on Friday, when ministers held talks at her country retreat of Chequers and emerged with a "collective" position on Brexit.
He flew out to Washington DC to convince Mr Trump that Iran would "go for a bomb" if he deserted the agreement, with his words falling on deaf ears as the U.S. later announced it would be pulling out of the deal.