Pro-Remain MPs rallied around Ms Davidson’s call for a rethink, prompting a rebuke for the Scottish leader from one of the Prime Minister’s allies.
Tories now face a difficult internal debate over whether to soften the strategy adopted by Mrs May, who attracted criticism for threatening to walk away from Brexit talks by insisting “no deal is better than a bad deal”.
It comes as Jeremy Corbyn confirmed Labour would stick to its plan for a “jobs first Brexit” prioritising EU workplace protections, but ending the free movement of people and exiting the European single market.
Ruth Davidson has pledged to flex her muscles after Tory gains in Scotland saved Mrs May from being forced from office, committing the 13 Conservative MPs in Scotland to “forcefully” make the case for an “Open Brexit” that puts trade and the economy ahead of cutting immigration.
A Scottish Conservative spokesman said Ms Davidson was not calling for the UK to remain in the single market, but Mrs May’s internal critics seized on the comments to demand a change of course.
Chancellor Philip Hammond is reported to have told the Prime Minister to put “jobs first” while negotiating a new deal with Brussels in a coded attack on her focus on controlling immigration.
And former Chancellor George Osborne, who was sacked by Mrs May and now is editor of the London Evening Standard, claimed there was no longer a majority in the Commons for a “hard Brexit”.
He suggested Ms Davidson would wield significant clout within her party as the “heroine” who prevented a Labour victory. “The interesting thing about her is she is now flexing her muscles and the most interesting thing she is flexing her muscles on is Brexit,” Mr Osborne said.
He told BBC One’s The Andrew Marr Show that leaving the EU without a deal would be “undeliverable” with the Conservatives relying on support from the DUP to government, “because they are absolutely committed not to have a hard border with the Republic of Ireland”.
Leading europhile backbencher Anna Soubry said the punishing election result meant the Mrs May “absolutely” had to water down her Brexit stance.
“She’s now got to make sure that she understands that British people have rejected a hard Brexit,” Ms Soubry said.
The former minister hailed Ms Davidson’s “mighty” breakthrough in Scotland and said that she believed calls for an open Brexit meant the UK to stay in the single market.
“I’ve listened to a lot of people, and the idea of a hard Brexit, the idea that no deal was better than a bad deal, people did not like that, which is one of the reasons we haven’t won this election.”
She said the Prime Minister “must not turn her back on British business, as I’m afraid she has”. “They want that single market, and they also want a proper immigration policy that recognises that we need immigrants, we need free movement, if businesses are to continue to flourish.”
But fellow Conservative MP Dominic Raab, who is widely tipped to return to government after a year on the backbenches, dismissed Ms Davidson’s comments, saying he did not recognise what she meant by an “Open Brexit”. “Like hard and soft Brexit, I’m not sure what Ruth means by that.
“She did a great job in Scotland but every Conservative Scottish, English and Welsh MP was elected on our manifesto, so obviously we deliver the plans in that manifesto as best we can – including, and especially, on Brexit.”
Brexit talks are due to begin in a week’s time.
Defence Secretary Michael Fallon played down the prospect of a major rethink on Brexit, but said there would be a renewed emphasis on the needs of business.
“We want to work with business on this. It is very, very important that we are careful about the existing trade we do with Europe.”