Theresa May has appointed an effective “deputy prime minister” as part of her fightback to stay in No 10 after her election disaster.

Long-time friend Damian Green was made First Secretary of State and Minister for the Cabinet Office, as the delayed Cabinet reshuffle got underway.

Mr Green has been shifted from work and pensions to the new role at the Prime Minister’s side, after she promised to work more openly with her party to rebuild her reputation.

Ms May and Mr Green – who backed Remain in the Brexit referendum – have known each other since they were Oxford students together in the 1970s and he is considered one of her few close allies.

Strikingly, his promotion comes hard-on-the-heels of the appointment of former MP Gavin Barwell, another ardent Remainer, as her new chief of staff.

Taken together, they will be seen as a strong signal that Ms May intends to reach out to the pro-Europe wing of her party – and even that she may step back from her planned hard Brexit.

However, the moves represent a gamble, because the Prime Minister has been strengthened since her election set-back by support from anti-EU MPs, who believe she is still the best bet to deliver full withdrawal on schedule.

Will Straw, the former Remain campaigner, tweeted: "Damian Green's a proper grown up who was a real asset to the Remain campaign board. Hope for a softer Brexit."

First Secretary of State is a title held by both William Hague and George Osborne under David Cameron, by Lord Mandelson when Gordon Brown was prime minister and by John Prescott under Tony Blair.

The move also marks a remarkable political recovery for Mr Green, who was once sacked by David Cameron as middle-ranking minister.

However, the last of three key Brexiteer Cabinet ministers, Liam Fox, will remain as Secretary of State for International Trade.

In a reshuffle of few changes, David Gauke was promoted from Treasury Chief Secretary to be the new Work and Pensions Secretary and David Lidington from Commons Leader to Justice Secretary.

But other key Cabinet ministers remain in post, including Jeremy Hunt (Health), Justine Greening (Education), Chris Grayling (Transport), Greg Clark (Business, Energy and Industrial Strategy). Sajid Javid (Local Government), Priti Patel (international Development), James Brokenshire (Northern Ireland Secretary) and Alan Cairns (Wales).

Elizabeth Truss was moved from Justice Secretary to be Chief Secretary to the Treasury, seen by some as a demotion because it is not a Cabinet post – although she will attend its meetings.

The reshuffle comes before Ms May’s showdown meeting with her own backbenchers, brought forward to 5pm on Monday.

Graham Brady, the chairman of the backbench 1922 Committee, said he did not want MPs to have to wait until Tuesday for the crucial inquest with the Prime Minister.

Backbenchers will demand to know why Downing Street announced that a deal was done for the Democratic Unionists to prop up the Conservatives in government – only to be forced to retract the claim overnight.

And they will want their fears of being tarred by association with the anti-gay marriage, anti-abortion party eased.