It comes with the embattled Prime Minister set to acknowledge that the loss of her Commons majority means she has to adopt a different approach to governing, signalling she is prepared to "debate and discuss" ideas with her opponents.
The paper also reported that a "kamikaze" group of right-wing Tory MPs said they could be willing to risk handing power to Jeremy Corbyn to kill off moves to reverse Brexit.
May called the election on June 8 in an attempt to cash in on high poll ratings and win support for her plan to make a clean break with the European Union in 2019.
Former chief whip Andrew Mitchell, who ran Mr Davis's unsuccessful 2005 leadership bid, sought to play down claims he told a private dinner that the PM had "lost her authority" and was "dead in the water", saying the account of the gathering was "overheated".
Rebel MPs from across Britain's political divide have formed a cross-party group to fight against the "hard and destructive" Brexit proposed by Prime Minister Theresa May.
In her speech, the PM will say that though the result of June's election was not what she wanted, "those defining beliefs remain, my commitment to change in Britain is undimmed".
"The Government also now have a wider choice, to be "timid or bold".
'We may not agree on everything, but through debate and discussion - the hallmarks of our Parliamentary democracy - ideas can be clarified and improved and a better way forward found'.
May's ability to carry on as prime minister and drive Brexit legislation through parliament with only a fragile majority behind her has been persistently questioned since the election.
"So I say to the other parties in the House of Commons: come forward with your own views and ideas about how we can tackle these challenges as a country".
He said: "Theresa May has finally come clean and accepted the Government has completely run out of ideas".
"In everything we do, we will act with an unshakable sense of objective to build the better, fairer Britain which we all want to see".
After the account emerged in the Mail on Sunday, Mitchell, a former development secretary, downplayed the remarks but did not explicitly deny having said them. But no-one will be fooled - the Tories are the party of the privileged few.
Some MPs said that a "brief dose of a Corbyn government" would "end in disaster" and boost the Tories in the long term.
"But there's big problems out there; the impact of digital technology on our economy, how we reform and fund our public services in the future, how we deal with a huge generational issue like social care".