Ian Blackford asked the Prime Minister on Wednesday to clarify if the messages were real following the publication of screenshots from his former ally Mr Cummings.
Speaking at Prime Minister’s Questions, the SNP Westminster leader asked Mr Johnson if they were genuine.
He said: "As we enter the chamber, we see what is reported to be WhatsApp communication between the Prime Minister and Dominic Cummings and perhaps the Prime Minister will clarify whether or not these are genuine comments he expressed on his health secretary?”.
The Prime Minister did not answer the question.
It follows the publication of messages appearing to show the Prime Minister labelling Mr Hancock as “totally f****** hopeless”.
Mr Cummings, who has blamed Mr Hancock for failings during the Covid response, published a screenshot apparently showing a WhatsApp exchange between himself and Mr Johnson.
In a blog post exceeding 7,000 words, Mr Cummings also published another private message about the struggles to procure ventilators for Covid-19 patients.
A contact appearing to be Mr Johnson replied on March 27, 2020: “It’s Hancock. He has been hopeless.”
In another message on April 27 last year, the Prime Minister appeared to call the situation around personal protective equipment (PPE) “a disaster” and suggested removing Mr Hancock.
He said: “I can’t think of anything except taking Hancock off and putting Gove on.”
The messages are Mr Cummings’s first attempt to publish supporting evidence since his select committee appearance where he accused the health secretary of lying, failing on care homes and “criminal, disgraceful behaviour”.
Mr Hancock has denied the Brexit campaigner’s allegations.
Justin Madders MP, Labour’s shadow health minister, responding to the publication of texts between Mr Johnson and Mr Cummings, said: “This is more evidence that the Conservatives were too slow to lockdown, too slow to deliver PPE and too slow to protect our care homes.
“With this evidence that even the PM thinks Hancock is useless, why in the worst pandemic in our history has he left him in charge?
“Hancock and Johnson need to respond to these latest revelations and immediately start the public inquiry into their handling of the pandemic.”
It came during a session that saw Mr Blackford also call for Mr Johnson to put his Australia deal to a Commons vote.
The SNP Westminster leader said: “In the Tories’ desperation to get a post-Brexit trade deal with somebody, with anybody, they’ve given the farm away, literally. It is blindingly obvious who are the winners and who are the losers in this deal.
“Australia’s economy will benefit to the tune of 1.3 billion dollars a year, the UK Government’s own assessment is the Australian deal is worth just – and I quote – 0.02 per cent of GDP.
"You would need 200 Australian deals to come close to mitigating the cost of Brexit.
“We were told that Brexit was all about taking back control, but for our farmers and for our crofters there’s been no scrutiny, there’s been no consultation and there’s been no consent.
"So if the Prime Minister is really confident about the benefits of this deal, does he have the guts to put it to a vote in this House?”
Mr Johnson replied: “The people of this country voted for this government to get on and deliver free trade deals around the world and I believe they were totally right.”
Mr Blackford also described the UK’s trade deal with Australia as “disastrous”, adding: “For all the spin, it’s clear that his Tory Government has just thrown Scottish farmers and crofters under their Brexit bus, just as they sold out our fishing community.”