Angela Rayner has handled the controversy over her living arrangements “in the right way”, Yvette Cooper has insisted after a former aide contradicted Rayner’s account.

The shadow home secretary said Rayner was “very keen to be able to provide the facts to the relevant authorities”.

Labour’s deputy leader is being investigated by Greater Manchester police (GMP) over whether she gave false information about her main residence a decade ago.

A former aide of Rayner’s, Matt Finnegan, disputed her account in a police statement on Sunday. He wrote that when he visited her in 2014 “there was no doubt in my mind” that she lived at the home that she maintains was her husband’s address, according to the Sunday Times.

GMP at first declined to investigate Rayner but reversed their decision on Friday after reassessing the information about the case.

The controversy relates to whether Rayner paid the right amount of tax on the 2015 sale of her council house on Vicarage Road in Stockport. She was registered to vote at the property and so did not have to pay capital gains tax, which would have been about £1,500.

Critics claim, however, that Rayner was living at her then husband’s property on Lowdnes Lane, about a mile away. There are also questions over whether she broke electoral law by registering to vote at the wrong address.

Rayner pledged to step down as deputy Labour leader if she was found to have committed a crime.

The police investigation follows a complaint by the Conservative MP James Daly. Labour has insisted that the controversy is politically motivated.

Cooper told the BBC’s Laura Kuenssberg on Sunday: “Angela welcomes the opportunity to be able to set out all the facts for the police, for HMRC, for the proper authorities, in the hope that that will then just draw a line under all of this.

“I think she has done the right thing by taking independent tax and legal advice and she has been clear throughout this that she has acted in accordance with that advice.”

Cooper confirmed that Keir Starmer had not seen the advice but that members of his team had. Starmer has faced questions about why he has not looked at the advice himself.

“I certainly wouldn’t expect to see the personal advice about personal finances of any of my colleagues,” Cooper told Sky’s Trevor Phillips on Sunday. “Keir has a very strong team around him, and rightly so – that’s why I think not only have they looked at this and been clear about it, but also Angela has as well.”

In his statement to the police, Finnegan said he visited Rayner around the time she became a parliamentary candidate at an address in Lowndes Lane in the summer of 2014, the Sunday Times reported. Finnegan reportedly wrote that there was “no doubt in my mind that this was Ms Rayner’s family home”.

Finnegan left Rayner’s employment with a £20,000 payout and non-disclosure agreement after accusing her of disability discrimination and unfair dismissal.

Some experts have argued that the police investigation is a waste of time and resources because the time limit for prosecuting potential offences has passed.

“We understand this is the run-up to local elections, we have seen this before, as we saw with the Durham case as well,” Cooper told Kuenssberg.

Durham police cleared Starmer and Rayner over allegations they broke coronavirus rules by having a takeaway curry with colleagues while campaigning for local elections in 2021.

The force initially investigated and found there was no evidence of rule-breaking, but looked at the case again following the intervention of the Tory MP for North West Durham, Richard Holden.