The Prime Minister is believed to have no option but to shake-up of her top team after Mr Green’s dramatic departure on Wednesday.
Sources claim Mrs May will use his resignation to reassert her authority as Brexit negotations hot up.
Brexit activists have urged her to sack pro-EU Cabinet members such as Chancellor Philip Hammond and Home Secretary Amber Rudd.
But Mrs May will likely be wary of upsetting the delicate balance between Brexiteers and Remainers in top Government roles.
Mr Green, a Remainer, was asked to step down after lying about claims pornography was found his work computer in 2008.
He became the third minister to leave the Cabinet in just six weeks, following the exits of Sir Michael Fallon and Priti Patel.
Brexit activists have urged the PM to sack Chancellor Philip Hammond
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Sir Michael, who voted Remain, was replaced by pro-EU Gavin Williamson as Defence Secretary.
And Ms Patel was replaced by a fellow Brexit campaigner, Penny Mourdant, as International Development Secretary.
Health Secretary Jeremy Hunt is “highly rated” and in line for a promotion, according to the Prime Minister’s allies.
One Cabinet member said: “It’s clear to everyone that Jeremy is on manoeuvres. He is deeply ambitious and seems convinced he has a good chance of greater things.”
Theresa May forced her deputy Damian Green to resign on Wednesday
Obviously these things are a matter for the Prime Minister
Mr Hunt, who campaigned for Remain, refused to rule himself out of the running to replace Mr Green.
But he insisted he was “passionate” about his current role, despite widespread anger at his management of the NHS.
He told the BBC yesterday: “Obviously these things are a matter for the Prime Minister, but as far as I am concerned I am a Health man.”
Mrs May is also expected to use the reshuffle to promote promising female MPs, according to the Financial Times.
Green was asked to step down after breaking the ministerial code
Mr Green’s exit from Cabinet was widely seen as a major blow to the Prime Minister’s authority.
As First Secretary of State and the de facto deputy PM, he was Mrs May closest political ally, standing in for her at the House of Commons.
He also managed relations with the DUP, which struck an informal coalition agreement with the Tories after June’s election.
Health Secretary Jeremy Hunt could get a promotion in the new year
Sources close to Mrs May said the decision to ask Mr Green to step down was made with “great sorrow”.
But they insisted the Prime Minister would head into 2018 in a relatively strong position.
One confidant said: “If you go back to where we were in September, when the party conference set was falling down around her, and where we are now, we are in an OK place.
“She has a strong team around her.”
Downing Street sources said the PM was in a relatively strong position
Another Downing Street insider said: “It’s better than it has been for a long time.”
A Number 10 spokesman refused to reveal when Mr Green’s replacement would be named.
He said: “In terms of roles, that’s obviously a decision for the Prime Minister, which she’ll announce in due course.”