MPs should not use their parliamentary office “for personal gain in any way”, the Health Secretary has said, following claims a former cabinet minister carried out his second job from Westminster.

Former attorney general Sir Geoffrey Cox could face an investigation by the Commons standards tsar over claims he broke code of conduct rules by using his parliamentary office to offer legal advice, in external work that is earning him hundreds of thousands of pounds on top of his £81,900 MP salary.

Sajid Javid said he did not want to comment on the specific allegations as he did not know the details, but said use of taxpayer-funded resources for private work “should not be happening”.

The Times reported that Sir Geoffrey, who is set to make more in 2021 than any other MP from outside earnings through his work as a practising barrister, used his Commons office in September to participate remotely to advise the British Virgin Islands over a corruption probe launched by the Foreign Office.

Deputy Labour leader Angela Rayner said the alleged use of the office appeared to be “an egregious, brazen breach of the rules” and has written to standards commissioner Kathryn Stone requesting a formal investigation.

Sir Geoffrey, as of Wednesday morning, does not feature on an online list of MPs alleged to have breached the code of conduct. An appearance on the list is the commissioner’s first step to confirm that a complaint has been received.

Mr Javid told BBC Radio 4’s Today programme: “As a general rule, no-one should be using things that are being funded by the taxpayer, whether that is your parliamentary office or whatever else, for your personal gain in any way.

“As a general rule – and again I’m not talking about an individual case – someone might take a phone call or might use their own phone for a video meeting or something like that, but I think as a general rule, that should not be happening.”

There have been calls for MPs to be prohibited from taking second jobs, but Mr Javid, who has formerly taken paid external work with US bank JP Morgan, admitted there could “perhaps be some” MPs who would quit Parliament if a ban was brought in.

It comes after Labour’s Ms Rayner told commissioner Ms Stone that Sir Geoffrey had “clearly broken” code of conduct rules which state that MPs should “not confer any… financial benefit” when using facilities paid for by public funds.

In a separate statement, she said: “This appears to be an egregious, brazen breach of the rules.

“A Conservative MP using a taxpayer-funded office in Parliament to work for a tax haven facing allegations of corruption is a slap in the face and an insult to British taxpayers.

“The Parliamentary Commissioner for Standards must investigate this, and the Prime Minister needs to explain why he has an MP in his parliamentary party that treats Parliament like a co-working space allowing him to get on with all of his other jobs instead of representing his constituents.”

In the British Virgin Islands commission of inquiry hearing on September 14, Sir Geoffrey can be heard in the online recording telling the commissioner: “Forgive my absence during some of the morning – I’m afraid the bell went off.”

The bell referred to could be the division bell that sounds off across the Parliament estate to alert MPs to a vote taking place.

Earlier in the proceedings, Sir Geoffrey appears to vacate his seat for about 20 minutes at around the two-hour mark in the video footage.

His Commons voting record shows that he voted in person on six occasions on September 14 to push through the Government’s health and social care levy.

The most recent register of financial interests showed that Torridge and West Devon MP Sir Geoffrey will earn more than £800,000 from Withers, an international law firm appointed by the British Virgin Islands (BVI) government in January.

The QC also disclosed in the register that from September 28 this year until further notice, he will be paid £400,000 a year by Withers for up to 41 hours of work per month.

The row over second jobs comes in the wake of a recommendation that former environment secretary Owen Paterson should be suspended for six weeks after the Commons Standards Committee found he had broken the centuries-old ban on paid lobbying by MPs.

In the bitter aftermath of the row, Mr Paterson announced he was quitting as MP for North Shropshire after 24 years, as an attempt by the Government to delay his punishment by ripping up the current standards system failed when opposition parties refused to offer their support.

Cabinet minister Mr Javid admitted he felt “regret” for supporting last week’s motion to change the system.

He told Sky News: “Look, I voted for that motion, I do regret it, because I think it was wrong for me to conflate the Owen Paterson case with the right of an appeal.”

The Health Secretary said he hoped the issue was “settled”, with the Government due to bring forward a motion to rescind the result and “accept the Owen Paterson report in full”.

Boris Johnson, who was previously well paid as a backbencher, including earning £275,000 a year for his weekly Daily Telegraph column, signalled that those in the Commons should focus on their electorates.

A spokesman for the Prime Minister on Tuesday said Mr Johnson thought an “MP’s primary job is and must be to serve their constituents and to represent their interests in Parliament”.

The PA news agency has contacted Sir Geoffrey’s office for comment.