The Observer and the BBC report the party is hoping to be granted an urgent question which would summon the Government to the Commons to account for allegations made by Mr Cummings.
In a blog post on Friday, Dominic Cummings accused Johnson of attempting an "unethical, foolish, possibly illegal" plan to get Conservative Party donors to fund a lavish refurbishment of the flat, which he lives in with fiancee Carrie Symonds and their son.
Truss declined to comment on Sunday when asked whether a donor initially paid for the refurbishment of the Downing Street apartment and Johnson then repaid the donor.
The Government has previously said Mr Johnson paid for the revamp - reported to have cost £200,000 - out of his own pocket.
British newspapers said on Sunday that Johnson's current advisers were anxious that Cummings would make further damaging allegations when he speaks to parliament next month about the government's handling of the coronavirus pandemic. What people want to know is that in line with the rules the prime minister has met the cost of this refurbishment.
Labour's shadow home office minister Jess Phillips described the public feud between Mr Johnson and Mr Cummings as "scrapping between two very powerful men who seem more interested in who's lying about what and who's leaking what than about the substance of the issue".
It follows Mr Cummings' dramatic departure a year ago from No 10 amid the fallout of a bitter internal power struggle with Ms Symonds.
Mr Case's office argues that the inquiry into who leaked plans for a second national lockdown in October is ongoing and nobody has been exonerated.
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According to Cummings, with regard to competence and integrity, the PM and his office had fallen below the appropriate standard.
Johnson's office has said the government had followed the rules over the refurbishment. "It matters because it's about integrity and it's about taypayers' money", Labour leader Keir Starmer said to BBC television.
The commission, which first raised the issue with the Conservative Party more than a month ago, confirmed at the weekend it was still looking into whether any of the sums relating to the work on the flat should have been declared.
"He said that it included "£58,000 to cover the payments the party has already made on behalf of the soon to be formed "Downing Street Trust" - of which I have been made chairman, as you know", he wrote.
The Daily Mail carried a claim that following the lockdown the Prime Minister had said he would rather see "bodies pile high in their thousands" than order a third one.
To date, no such trust has been formed.
"The party is working with us on this".