Downing Street has argued that Kathryn Stone, the parliamentary commissioner for standards, should not investigate who paid for the renovation of Boris Johnson’s flat, in the latest apparent attempt to exert pressure on the independent regulator. Boris Johnson’s spokesperson said there was no need for Stone to look into the matter as the No 10 flat was not connected to the prime minister’s role as an MP, prompting Labour to warn that it was vital the matter was properly scrutinised. “It is not for the prime minister or cabinet ministers to decide what the independent anti-corruption commissioner investigates,” said Angela Rayner, Labour’s deputy leader. Meanwhile, Johnson said it would be up to a future cross-party process led in part by the Commons Speaker, Lindsay Hoyle, to determine whether MPs should have second jobs, in the wake of the resignation of the Tory MP Owen Paterson. Speaking on a visit to Northumberland, Johnson said he hoped there would be “cross-party agreement on a way forward, including an appeals process for very difficult and very sad cases such as the ones we’ve seen”. Asked if MPs should be able to have second jobs, after Paterson was found to have breached lobbying rules in working for two companies, Johnson said: “All those kinds of things are issues that the Speaker’s panel – or whatever he sets up – are going to have a look at.” Hoyle is expected to set out his plans before an emergency Commons debate on the issue on Monday afternoon. Johnson will miss the debate, as No 10 said he was taking the train to and from Northumberland rather than flying. Last week Johnson decided not to declare the cost of a free holiday in Spain at a luxury villa owned by the family of Zac Goldsmith, the former Tory MP whom Johnson made a peer and junior minister. Johnson listed the near week-long stay in the Marbella property in October in the register of ministerial interests, But on Friday No 10 said that as the villa was provided by another minister, he did not also need to declare it in the separate register of MPs’ interests. Unlike the ministerial equivalent, this gives monetary values for gifts and donations. On Sunday the environment secretary, George Eustice, argued that the refurbishment of Johnson’s No 10 flat – for which Johnson was loaned at least £58,000 from Conservative party funds – was also “clearly a ministerial issue”. Johnson’s spokesperson, when asked if the prime minister believed Stone should be able to investigate the flat refurbishment, said: “Obviously it’s a matter for her on that. The interest, as you know, has been transparently declared by the prime minister following advice from Lord Geidt, the independent adviser. “And the Commons rulebook is very clear that such ministerial code declarations do not need to be double-declared. And the flat was clearly a ministerial matter, as the PM only occupies it by virtue of his office.” The spokesperson reiterated that the cost of the Spanish holiday would not be formally declared. “It’s for the commissioner to decide and set out her rationale. That holiday has been declared in the appropriate manner, as we set out last week,” he said. Stone has been placed under considerable government pressure in recent days. After her ruling that Paterson had broken rules on paid lobbying was overturned in a government-led Commons vote, the business secretary, Kwasi Kwarteng, indicated that she should step down. Labour has called for an investigation into Kwarteng’s comments. When asked on Monday whether she backed Kwarteng’s view, the international trade secretary, Anne-Marie Trevelyan, said Stone “needs to be allowed to get on with her job”.