Theresa May is offering to hold talks with Speaker John Bercow on overhauling Commons disciplinary procedures amid mounting reports of abusive and inappropriate behaviour towards women.

A list of 13 MPs facing harassment allegations has been circulating at Westminster, according to The Daily Telegraph, as Number 10 again made clear any unwanted sexual behaviour was “completely unacceptable”.

Meanwhile the Guido Fawkes website claimed Tory aides had compiled a spreadsheet of 36 Conservative MPs – including 20 ministers – accused of inappropriate behaviour. The Conservatives declined to comment.

Over the weekend the Prime Minister ordered a Cabinet Office inquiry into whether International Trade Minister Mark Garnier had breached the ministerial code over claims he asked his Commons secretary to buy sex toys and called her “sugar tits”.

Mrs May was also facing calls to suspend a second senior Conservative, former Cabinet minister Stephen Crabb, after he was reported to have admitted sending explicit messages to a 19-year-old woman he interviewed for a job.

In a letter to Mr Bercow, the Prime Minister said the current grievance system for dealing with complaints by MPs’ staff lacked “teeth” as there was no contractual requirement for MPs to follow its procedures.

Stephen CrabbTheresa May is coming under pressure to suspend Stephen Crabb (Dominic Lipinski/PA)

“I do not believe that this situation can be tolerated any longer. It is simply not fair on staff, many of whom are young and in their first job post-education,” she wrote.

“It is vital that the staff and the public have confidence in Parliament and resolving this employment irregularity on a cross-party basis can play an important role in this.

“I would be grateful if you would be able to use your office to assist me in doing all we can to ensure that the reputation of Parliament is not damaged further by allegations of impropriety.”

Liberal Democrat leader Sir Vince Cable signalled his support for Mrs May’s initiative, saying: “Parliament clearly needs improved procedures to respond to allegations of harassment.”

The former chairman of the Committee on Standards in Public Life, Sir Alistair Graham, also backed the move but warned against making the system too complicated, and questioned Mrs May’s suggestion of a mediation system.

“This could be a turning point,” he told BBC Radio 4’s Westminster Hour.

“But the danger is you just get an accretion of more and more systems which just makes the whole arrangement unworkable – which is why I’m a great believer in simplicity, clarity, and making sure that when complaints are made they’re investigated very quickly.”

Labour MP Jess Phillips called on Mrs May to suspend Mr Crabb from the Conservative Party pending an investigation into his conduct.

Writing in the Times Red Box email, she said: “Theresa May must put her money where her mouth is and investigate Stephen Crabb while his whip is removed.”

Tory former minister Anna Soubry, who has asked Mr Bercow to grant an urgent question on the issue in the Commons, said any new grievance procedure must protect "all workers in the Palace of Westminster" not just MPs' staff.

Many MPs were "fed up to the back teeth with the level of misogynistic abuse" they face, she told BBC Radio 4's Today programme, adding: "There is a problem and we are all responsible for sorting it out.

"Theresa May, because she is a woman leader of a party, absolutely will - I'm confident of that."

Labour MP Lucy Powell compared the situation with the Hollywood abuse scandal, with people desperate to advance in an industry.

She said: "When you have that mix of lots of desperate people in that environment, this sort of power abuse - because that's what it is, it's about a power inequality - can thrive."

But she said Mrs May was "too weak" to tackle the problem and in Mr Garnier's case "referring something off to the ministerial code is just not sufficient".

Labour's John Mann said he knew of four cases of "objectionable behaviour", including two where women made complaints, one to her political party and one to the parliamentary authorities.

One of the allegations "passes the criminal threshold" and one is "appalling", he said.

Mr Mann told LBC: "These are within the last couple of years, both of these, so we're talking of actual complaints where the woman have, in one case, been told to get lost, it's nothing to do with us, and in the other case, nothing is done and she's not even got back to with a response, which shows how badly actual complaints are treated and these are serious complaints."

The MP said he was calling for the cases, which relate to two different parties but do not involve Conservatives, to be reopened.

He added: "In one case, the police were involved and then the police couldn't prosecute because the alleged assault took place abroad."