Downing Street has refused to back calls for Sir Philip Green to be stripped of his knighthood over sexual harassment and bullying allegations, saying it is a matter for independent review.

After Peter Hain used parliamentary privilege in the House of Lords to name Green as the mystery figure at the centre of the allegations, MPs and others called for the title to be removed. However, a spokesperson said Downing Street did not have the power to make that decision.

The spokesperson said on Friday that the independent body that oversaw such issues was “constantly reviewing evidence in relation to matters like this”, adding that the prime minister had been clear that “bullying and sexual harassment in the workplace is against the law and that kind of behaviour can’t be tolerated”.

The Liberal Democrat leader, Vince Cable, was among the MPs who called for Green to lose his honour. “He narrowly and luckily escaped losing his knighthood over the pensions scandal,” he said, referring to the House of Commons’ unusual move two years ago to back a motion calling for his knighthood to be removed over the collapse of BHS. “If these allegations are correct, he should certainly be stripped of his knighthood.”

Other MPs backing the call included Frank Field and Jess Phillips.

The mayor of London Sadiq Khan joined calls for Green to be stripped of his honour should it be proven that the businessman signed off on non-disclosure agreements to stifle allegations of bullying and sexual harassment. Speaking in Brussels, Khan said: “You give knighthoods to those who have served our community and society. But I think just in the same way you reward good behaviour, I think you should punish bad behaviour”.

Among those to have lost their honours in the past are the former RBS chief executive Fred Goodwin and the former Zimbabwean president Robert Mugabe.

All such decisions are made by the independent honours forfeiture committee – an ad hoc body led by a senior civil servant. They have to conclude that an individual is “bringing the honours system into disrepute” before removing any titles.

Lord Hain defended his decision to name Green on Thursday. The former Labour cabinet minister said his decision to use parliamentary privilege to circumvent the courts was “the right thing to do”.

“I considered it extremely seriously before I said it,” said Hain, who has been criticised by some lawyers for his action. “I’m discharging my function as a parliamentarian. What concerned me about this case was wealth and the power that comes with it, and abuse.”

Hain told BBC Two’s Newsnight: “It’s for members of the public and others to judge whether I was right or wrong. There’s no point in being there if you never discharge that and deploy the precious right of parliamentary privilege, to be used extremely carefully with integrity and very responsibly.”

Hugh Tomlinson QC, the founder of the Hacked Off campaign for press reform, told BBC Radio 4 the decision to name Green in defiance of the courts was “completely improper”, given the injunction was only a temporary restriction pending a full trial on whether the Daily Telegraph’s story was in the public interest.

“Parliament can’t trespass into areas of the courts and say we think the courts have got it wrong,” he said. “That’s what Lord Hain is effectively doing.”

Dominic Grieve, the Conservative MP and former attorney general, also criticised Hain.

“It was an entirely arrogant decision that had absolutely no regard for judicial process or the rule of law,” he told the BBC’s World at One. “Parliamentary privilege is very important, but like any power which is extremely important it is open to abuse. I can’t see - looking at this particular matter - that Peter Hain can argue that he hasn’t abused it.”

Green was named after the Telegraph said it had been blocked by the courts from reporting that he was the subject of allegations of “discreditable conduct” made by five employees, which the newspaper claims include allegations of sexual harassment, bullying and racial abuse.

The newspaper said the allegations that Green had used non-disclosure agreements to limit coverage would “reignite the #MeToo movement against the mistreatment of women, minorities and others by powerful employers”.

Green, who is thought to have spent hundreds of thousands securing the injunction, denied allegations of wrongdoing, saying: “To the extent that it is suggested that I have been guilty of unlawful sexual or racist behaviour, I categorically and wholly deny these allegations.”

Courts have no ability to restrict reporting on the activities of parliament, meaning that if an MP or peer speaks in the chamber then the media can report it. Politicians have previously used this right to undermine reporting restrictions on the oil trading business Trafigura and identify the footballer Ryan Giggs, who had taken out a superinjunction.

Green was knighted in 2006 by Tony Blair’s Labour government for “services to the retail industry”. But his honour has come under scrutiny over his tax arrangements – many of the businesses he runs are owned by his Monaco-based wife, Tina – and his decision to sell British Home Stores shortly before it collapsed with a substantial pension deficit.