Team Ruth set to meet at Westminster with pledge to put Scotland's case forcefully

A senior Tory source said: “The boost in Scottish Conservative numbers will extend Ruth and David’s influence. It will put a greater Scottish perspective on things.”

Ms Davidson, who supported Remain during the In/Out referendum, has already insisted there needs to be wider consensus on the terms of leaving the EU and has made clear she will seek to use her enhanced influence to push for an “open Brexit”. She described this as about ensuring that “we increase freedoms rather than increase barriers; it is about making sure that put free trade at the heart of what it is we seek to achieve as we leave".

However, there are divisions on Brexit within Scottish Tory ranks. Ross Thomson, the new Tory MP for Aberdeen South, was a prominent Conservative spokesman for the Leave campaign in Scotland.

The scale of the Scottish party's success in the election, including the toppling by the Tories of two of the SNP’s most senior figures at Westminster - Alex Salmond, the former First Minister, and Angus Robertson, the deputy leader - has greatly increased Ms Davidson’s influence inside the Conservative Party at a UK level.

Within hours of the result, the MSP’s enhanced authority was on show when she made public that she had sought a guarantee from Mrs May that the proposed Conservative deal with Northern Ireland’s Democratic Unionist Party would not lead to any retreat on gay rights or equal marriage policy.

Ms Davidson’s influence is only likely to grow in the coming weeks and months. Already able to attend cabinet meetings, it is expected that she will visit Westminster more regularly, which will increase speculation that, at some point in the future, she might seek to create a permanent career in London rather than in Edinburgh.

She said at the weekend: "I'm going to make sure that along with the 13 MPs we're sending to Westminster, we put Scotland's case forcefully and make sure that we deliver for Scotland and make sure that the Union delivers for Scotland too."

It is thought that the Scottish Conservative Westminster Group – larger than the Democratic Unionists - will vote as a discrete bloc but not seek to overturn policies in the UK party’s manifesto and will not attempt to scrap the so-called “rape clause,” which put the party and Ms Davidson in particular under pressure during the election campaign, as the Tories believe the two-child welfare cap is popular with voters.

However, the emphasis will be on policies going forward covering such issues as fishing, farming and the new borderlands growth deal; the Scottish Tories won all three borders seats.

The new group met in Stirling at the weekend to discuss how to approach policies in a new political landscape with the UK Tories running a minority administration propped up by Arlene Foster's DUP.

Sources stressed that while Scottish Tories would operate as a distinctive group, there was no intention to seek to defeat Mrs May on any key votes. John Lamont, the new MP for or Berwickshire, Roxburgh and Selkirk, who has been the party’s chief whip at Holyrood is expected to take up the same role in the Scottish group at Westminster.

The larger than expected Scottish Tory group in London is likely to mean that Mr Mundell will get a new minister at the Scotland Office.

Lord Dunlop has resigned as the party’s spokesman in the Lords. Speculation is rising that Baroness Goldie, currently a whip in the second chamber, could take over the role.

The increase to 13 Scottish Tory MPs means that Mr Mundell, the sole Conservative from north of the border in the Commons since 2005, means he will no longer carry the soubriquet of the "UK's most expensive MP".

Meantime, Ms Davidson confirmed she has given her "full support" to Mrs May staying on as PM.

"The Prime Minister was the leader that made sure that we came out with the largest number of votes and seats, she has my full support as she makes sure that we move forward with a strong government," she said.