The remarks were said to have been made last autumn, just as England went into a second lockdown.
The PM has strongly denied saying the phrase, describing the reports as "total rubbish".
Labour's Rachel Reeves urged Mr Johnson to apologise.
She called the comments "stomach-churning".
The government is also facing questions over possible donations made to re-decorate the prime minister's flat, and an investigation into leaked information about England's second lockdown.
Mr Johnson's comments - first reported in the Daily Mail - came at the end of October when the government announced there would be a second lockdown in England following a sharp rise in coronavirus cases.
BBC political editor Laura Kuenssberg says, at the time, the prime minister was reported to have had big concerns about the implications of another lockdown on the economy and non-Covid related health issues.
"This does take us back to that moment and back to the very serious claims made by some people who were involved in the decision making - including some ministers - that the hesitancy around the second lockdown did cost lives," she said.
Asked about the comments earlier on Monday, Mr Johnson denied having made them adding that the public wanted the government "to make sure that the lockdowns work, and they have".
The PM's spokesman said the reported comments were false adding: "This is untrue and the PM has denied it… I'm not aware of anyone else making that statement."
Mr Johnson was also defended by Cabinet Office minister Michael Gove who told MPs he was in the room during conversations about the lockdown but "never heard language of that kind".
Labour leader Sir Keir Starmer said he was "astonished" by the reports adding: "Everybody would be deeply concerned, not least all those families who have lost someone in the pandemic."
The prime minister was also strongly criticised by other opposition MPs in the House of Commons.
Ms Reeves - Labour's shadow Cabinet Office minister - said the prime minister was a man who "would rather the bodies pile high than act on scientific advice - but they are not bodies; they are people, they are loved ones and they are deeply missed."
She called for an urgent public inquiry into the government's handling of the pandemic adding: "This is all about conduct, character and decency - frankly, our country deserves an awful lot better than this."
The SNP's Alison Thewliss said the comments were "not befitting the office of prime minister" describing his words as "despicable, cruel and callous".
Matt Fowler, co-founder of the group Covid-19 Bereaved Families for Justice, which is calling an inquiry into the pandemic, said the comments had caused outrage among members.
He said: "These callous comments will have caused untold hurt to hundreds of thousands of us across the whole of the country.
"This demonstrates exactly why an urgent inquiry is so vital, to understand the decisions, rationale and consideration in protecting our loved ones that the government chose."
This comes amid a bitter row between Downing Street and the prime minister's former senior adviser Dominic Cummings.
Last week, Mr Cummings published a blog in which he:
* denied leaking text messages sent between Mr Johnson and businessman Sir James Dyson
* accused the PM of planning to have donors "secretly pay" for the refurbishment of his flat
* denied leaking details of November's second coronavirus lockdown in England
* claimed Cabinet Secretary Simon Case had cleared Mr Cummings of being the source of the leak
* alleged Mr Johnson had considered trying to block an inquiry into the leak in case it involved a friend of his fiancee Carrie Symonds.
Earlier, Mr Case gave evidence at an MPs' committee and said the inquiry into the second lockdown leak is ongoing; however, he added that "given the time that has now passed, I think it's probable the team will not successfully identify the source or sources".
A Cabinet Office source told the BBC that no-one had yet been exonerated in the investigation.
Asked if he would launch an inquiry into the leak of the prime minister's lockdown comments, Mr Case said he would have to discuss that with Mr Johnson.
The prime minister is also facing questions about the funding of his Downing Street flat refurbishment.
Over the weekend, ministers had said the prime minister "personally paid the bill" for the flat but did not answer questions on whether a Tory party donor initially provided the money to him.
Mr Johnson has said he would make any necessary declarations about donations "in due course" and Mr Case - who is the UK's top civil servant - said he would review how the refurbishment was paid for.
Speaking in Wrexham, Mr Johnson said the public were more interested in what the government was doing to move the country "cautiously but irreversibly through the steps of the road map to unlock and to get our country going".
He said the disease was "under control" and that deaths and hospitalisations were low but added "that doesn't mean that we've got it totally licked".
"We've got to be realistic about that; unfortunately, there probably will be another wave of the disease."