Meg Hillier, the Labour chair of the public accounts committee, said Durham Police and the Met should look again at the case after two people who claimed they saw Mr Cummings in Durham on 19 April complained to the police watchdog.
Boris Johnson told Ms Hillier in April that he had seen evidence to prove his chief adviser was not in Durham on that date but refused to make such evidence public.
Mr Cummings previously admitted to a trip to Durham during lockdown in late March, which he claimed did not break coronavirus restrictions, but denied allegations of a second trip and said he had evidence to prove he was in London on that day.
“If the PM just publishes the evidence he’s seen there would be no need for police time to be spent on this – the PM should put truth and public trust first and publish what he’s seen,” Ms Hillier told The Guardian.
In a letter seen by The Daily Mirror, the Labour MP told Mr Johnson that the issue went “right to the heart of trust in the government”.
She wrote: “You spent six hours of your time discussing this with Mr Cummings at the height of the pandemic.
“You spent 90 minutes with senior MPs asking you questions on behalf of the public, now I urge you to take a few minutes to publish the evidence you have seen.”
Angela Rayner, Labour’s deputy leader, has also called on Downing Street to produce its evidence for Mr Cummings’ denial.
A Downing Street spokesperson said on Thursday that the prime minister believed his adviser had “behaved reasonably” and considered the matter closed.
Earlier this year, a Durham Police investigation concluded there was “insufficient evidence” to support the allegations of a second trip by Mr Cummings in April.
It came as researchers said Mr Johnson’s handling of the controversy had “negative and lasting consequences” for the public’s trust in the government’s response to the coronavirus pandemic.
The authors of a study published in the Lancet medical journal found a clear decrease in public confidence in the government when the story broke in May, with trust continuing to fall sharply in the days that followed.
The research, which analysed data from the UCL Covid-19 Social Study, showed public trust never recovered following the prime minister’s refusal to sack Mr Cummings.