The Scottish Conservative leader said Mrs May had “had a setback” in Salzburg after European Council president Donald Tusk said the other 27 European Union members wanted her Chequers proposals to be redrawn.
But she paid tribute to the “remarkable and astonishing” resilience of the Prime Minister, and backed her to reach an agreement that would avoid the UK crashing out of Europe without a deal.
Ms Davidson, speaking at an event in Edinburgh organised by The Times Scotland, said she had always “expected people to negotiate hard” as Britain prepares to leave the European Union.
She stated: “Of course it is concerning that we are still in a place where it is not clear to everybody what is going to happen and when and how it is going to happen. But I still think there will be a deal
“I would really quite like some of my colleagues in Westminster to shut up for a bit. Put their sharp elbows and their personal ambition away and just let the Prime Minister do her job and go into bat for the country and bring home a deal.
“That’s what I wish more than anything for the next few weeks and months, genuinely,”
Mrs May’s Brexit proposals have come under fire from hardline Brexiteers within the Conservatives, such as former foreign secretary Boris Johnson.
But with time running out for the UK to reach a deal, Ms Davidson said: “It is absolutely right that people have their say about what sort of Brexit they want, but we are right down to crunch time now, we’re talking weeks not months.
“We’re sending her into negotiate, sitting across the table from 27 other nations, we’re asking her to bring something back for the country.
“I don’t care what you want your next job to be. Your job now is to get behind and let her deliver for the country. Let the woman do her job.”
Speaking about the talks with other European leaders in Salzburg, Ms Davidson said: “Today she has had a setback.
“But she will get up tomorrow morning and she will keep going, and I think the reserves that she has - she never talks about being type 1 diabetic, she never talks about now being over 60 or any of that sort of stuff - but the resilience she is showing under the most extreme pressure, we haven’t seen in politics since the financial crash of 2008 and before then for a couple of decades before that. It is genuinely remarkable and astonishing.”
She stated: “It is in everybody’s interests, the United Kingdom’s and the EU’s, to make sure there is a deal. I still believe there is political will on both sides.”