The Prime Minister is giving her government a New Year revamp after close ally and de-facto deputy Damian Green was sacked over allegations he had porn on his computer.
Reports suggest that around half a dozen of her senior ministers could be axed or moved.
Mr Hunt is set to replace Mr Green as First Secretary of State, with ex-nurse Anne Milton to replace him as health secretary.
The only potential barrier to that move is Mrs May may find it difficult to move him in the midst of an NHS winter crisis which has seen tens of thousands of operations cancelled.
Another name in the frame for the position is Transport Secretary Chris Grayling who was one of the first pro-Brexit ministers to back Mrs May when she stood for the Conservative leadership in 2016.
PA
The most senior members of the Government – Chancellor Philip Hammond, Foreign Secretary Boris Johnson, Home Secretary Amber Rudd and Brexit Secretary David Davis – will all remain in their present posts.
However Education Secretary Justine Greening, Conservative Party chairman Sir Patrick McLoughlin, Business Secretary Greg Clark and the Leader of the Commons Andrea Leadsom are all reported to be vulnerable as Mrs May seeks to assert her authority.
Sir Patrick looks set to pay the price for the party's failure at last year's snap general election when the Tories saw their Commons majority wiped out.
It is thought that Mrs May will take the opportunity to bring forward some more junior ministers, with Immigration Minister Brandon Lewis and Justice Minister Dominic Raab among those tipped for promotion.
The Daily Telegraph reported she will appoint a "minister for no deal" specifically to prepare for the eventuality that the UK leaves the EU without an agreement, with the post likely to go to Steve Baker – a prominent Leave campaigner who is already a Brexit minister.
Conservative Party's leadership contenders
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“I'm not a quitter. I'm in this for the long term”
Prime Minister Theresa May
The Prime Minister is also believed to want more women and MPs from ethnic minorities in her team with Suella Fernandes, the leader of the backbench Eurosceptics, Seema Kennedy and Rishi Sunak among those who could enter the Government for the first time.
Overall the reshuffle will be more extensive than that carried out by Mrs May following her humiliation in last year's general election when she felt able to make only limited changes to the Cabinet.
Speaking at the weekend, Mrs May insisted she still wanted to lead her party into the next election due to take place in 2022, declaring: "I'm not a quitter. I'm in this for the long term."
But pressed on whether she would still be there the next time the country goes to the polls, she appeared to acknowledge that the decision may not be entirely down to her, saying: "Obviously I serve as long as people want me to serve."