The prime minister's decision comes amid severe winter pressures on the NHS which have seen thousands of operations postponed.
It is understood that Mrs May does not intend to appoint a First Secretary of State in what is expected to be her biggest reshuffle since taking office in 2016.
May's relaunch got off to a chaotic start after the Conservative Party accidentally tweeted that Chris Grayling was the new party chairman when the job had actually been given to Brandon Lewis.
Mary Bousted, joint general secretary of the NEU teaching union, said: "Just when positive things are starting to happen, to have her moved and the Department for Education in turmoil again would not be good for education".
Ms Greening said: "Social mobility matters to me and our country more than a ministerial career".
But the former justice secretary has not been given the title of First Secretary of State, which marked Mr Green out as Theresa May's effective deputy.
And Amber Rudd was confirmed in her position as Home Secretary, as one of a group of the most senior ministers - also including Chancellor Philip Hammond, Foreign Secretary Boris Johnson and Brexit Secretary David Davis - expected to stay in their current posts.
Sajid Javid has kept his job as communities secretary in a renamed Ministry of Housing, Communities and Local Government.
At the end of the meeting, Mr Hunt remained as Health Secretary, but had responsibility for social care added to his brief.
As well as McLoughlin, education secretary Justine Greening, business secretary Greg Clark and Commons leader Andrea Leadsom are all reportedly vulnerable.
And a backlash grew after Lewes MP Maria Caulfield was appointed the party's vice-chair for women - despite having led opposition to the decriminalisation of abortion.