Jacob Rees-Mogg has admitted a Covid-19 test for his son was couriered to his home during a national shortage.
He accepted it might be deemed "special treatment" but defended the arrangement due to his workload as a cabinet minister at the time.
WhatsApp messages leaked to the Telegraph appear to show former health secretary Matt Hancock's adviser helped send the test in September 2020.
The Telegraph has obtained more than 100,000 of Mr Hancock's messages.
The paper has published messages it says were sent on 10 September, 2020 by Allan Nixon, Mr Hancock's political special adviser at the time.
One suggests an initial test for one of Mr Rees-Mogg's children had been lost by the laboratory.
It goes on to say "we've got a courier going to their family home tonight, child will take the test, and courier will take it straight to the lab. Should have result tomorrow am".
Responding on his GB News chat show on Wednesday, Mr Rees-Mogg said "one of my children needed a test and that put everybody into quarantine".
When he was informed his child's initial test had been lost, "it was raised with the department of health and they decided to send me a test to a member of my family", he said.
"So if I received special treatment, it wasn't because I had requested it, but actually it allowed a government minister to get back to work with a child who didn't have Covid
in the first place," the former business secretary said.
He said the Covid
testing system had not been working but "I accept it wasn't working for other people too".
Labour has urged the government to determine whether Mr Rees-Mogg received special treatment and not to "hide" behind the Covid
A spokesman for Labour leader Sir Keir Starmer said: "The government is more than capable of answering those questions now and ensuring that we can have clarity and transparency on those issues."
The Liberal Democrats have written to Prime Minister Rishi Sunak to ask him what he knew and when regarding claims that ministers had access to "priority testing".
The party's deputy leader Daisy Cooper said: "These reports are just more evidence that it's one rule for Conservative ministers and another for everyone else.
"The government must urgently publish exactly how many Conservative ministers, MPs and their families had access to priority testing when the public faced a national test shortage. The public deserves to know the truth."
During Prime Minister's Questions on Wednesday, Mr Sunak said the inquiry was "the right way for these things to be looked at" and that he would not comment on "piecemeal bits of information".
Around the time of the Rees-Mogg messages, Sarah Marsh, director of testing at NHS Test and Trace, tweeted "heartfelt apologies to anyone who cannot get a Covid
test at present".
In the same month, media reports said government testing laboratories across the UK were facing a backlog of 185,000 Covid
tests. Under the rules at the time, people had to isolate until a negative test was recorded.
Mr Hancock said it could take weeks to resolve issues around Covid
testing and admitted there were "operational challenges" related to a surge in demand.
The Telegraph has been handed more than 100,000 WhatsApp messages linked to Mr Hancock's time as health secretary at the height of the pandemic.
The text messages were passed to the newspaper by journalist Isabel Oakeshott, who has been critical of lockdowns. Ms Oakeshott was given copies of the texts while helping Mr Hancock write his book, Pandemic Diaries.
The BBC has not seen or independently verified the WhatsApp messages nor the context in which they were sent.
Mr Hancock's spokesperson said the Telegraph had published "partial leaks" that presented a "distorted account of the pandemic" designed to "fit an anti-lockdown agenda".
The spokesperson said the messages had been made available to the public inquiry into the government's response to the pandemic.
"Instead of spinning and leaks we need the full, comprehensive inquiry, to ensure we are as well prepared as we can be for the next pandemic, whenever it comes," the spokesperson said.