Meanwhile, Labour is maintaining pressure over the funding of refurbishments to the prime minister's Downing Street flat, saying questions remain over how the work was paid for.
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.
On Friday, the Prime Minister denied trying to block the leak inquiry, saying the public could not "give a monkey's" about such matters.
Dominic Cummings, who left Johnson's staff suddenly late past year having been his most influential adviser on Brexit and the 2019 election campaign, said in a blog on Friday he had told Johnson he thought his plans to have donors pay for the renovation were "unethical, foolish, possibly illegal".
The opposition will ask Commons Speaker Sir Lindsay Hoyle to grant an urgent question requiring a minister to come to the House to respond to the charge that Mr Johnson plotted for Tory donors to secretly fund the work.
Meanwhile, asked about the claim that Mr Johnson tried to stop a leak inquiry, Ms Truss dismissed the allegation as "tittle-tattle that's being promoted and don't think it really addresses the key issues that people in Britain really care about".
Lord Barwell, who was Theresa May's chief of staff when she was prime minister, said he was "surprised" at how long the inquiry into the leak was taking.
Trade Secretary Liz Truss, told the BBC's Andrew Marr Show on Sunday that Mr Johnson covered the renovation costs for his Downing Street flat "from his own pocket".
"I absolutely believe and trust that the prime minister has done that".
"All the details will be given out in line with the rules", Ms Truss added.
However, Ms Truss repeatedly refused to say when Mr Johnson paid the costs, amid suggestions a Conservative donor had initially offered up a £58,000 sum. "There are certain reporting procedures and those procedures are being followed".
Ms Truss stressed the so-called "chatty rat" leak inquiry was still "ongoing" - almost six months after it was first launched - adding: "I hope it finds who the individual was who conducted this leak because it's very, very unhelpful when confidential is leaked from the government".
That counters Mr Cummings' claim that he was cleared by Mr Case of being the source - described as the "chatty rat" - who told reporters about the imminent lockdown.
Labour has also called for a new independent commission to be launched, on ethics and standards in government.
"Whether I back Dominic Cummings's view or Boris Johnson's view, what we need is a proper independent inquiry where it isn't about two boys fighting, it's about taxpayers in our country".
But she added there was also a need for an independent inquiry.