The report comes as more than 100,000 WhatsApp messages were leaked to The Daily Telegraph by journalist Isabel Oakeshott, who worked with Mr Hancock on his memoir The Pandemic Diaries.
Sir Gavin also reportedly said in the messages that teaching unions “just hate work” - sparking outrage from a teaching union boss on Thursday.
In the latest exchange, released by the Telegraph on Thursday, Mr Hancock allegedly called teaching unions “a bunch of absolute arses”, to which Sir Gavin responded: “I know they really really do just hate work”.
In another exchange, reportedly made in May 2020 as schools prepared to reopen to pupils following months of lockdown, the two ministers appeared to discuss the supply of personal protective equipment (PPE) to schools.
The UK was experiencing a shortage of PPE at the time, and Sir Gavin appeared to ask Mr Hancock to help make it possible for schools to access supplies through local resilience forums - multi-agency partnerships made up of local public services including the NHS, emergency services and local authorities.
Referring to school staff, Sir Gavin reportedly responded: “Some will just want to say they can’t [access PPE] so they have an excuse to avoid having to teach, what joys!!!”
Downing Street on Thursday afternoon said it does not agree with any suggestion that teachers were trying to find an “excuse” not to work during the Covid-19 pandemic.
The Prime Minister’s official spokesman said Rishi Sunak “hugely values our hard-working teachers who did so much during the pandemic to help minimise disruption to children’s education”.
Geoff Barton, general secretary of the Association of School and College Leaders union, has condemned the ministers’ reported comments on teaching unions.
“It’s contemptible because we have to remind ourselves that this was an age of extraordinary anxiety,” Mr Barton told BBC Breakfast on Thursday.
“We hadn’t got vaccines.
“And the Government was starting to look to the teaching profession to welcome those young people back into school. It was a huge debate going on, very snarky debate about whether face coverings should or shouldn’t be worn.
“And essentially, the very people who then brought those young people back into school are being described in those snide terms by the former education secretary.”
Mr Hancock’s team has hit back at Ms Oakeshott and The Telegraph for publishing the messages, saying the story is “wrong, based on partial, spun leaks,” and that they failed to approach Mr Hancock before publication.
In a statement issued on Thursday, Matt Hancock said: “I am hugely disappointed and sad at the massive betrayal and breach of trust by Isabel Isabel Oakeshott. I am also sorry for the impact on the very many people – political colleagues, civil servants and friends – who worked hard with me to get through the pandemic and save lives.
“There is absolutely no public interest case for this huge breach. All the materials for the book have already been made available to the Inquiry, which is the right, and only, place for everything to be considered properly and the right lessons to be learned. As we have seen, releasing them in this way gives a partial, biased account to suit an anti-lockdown agenda.
“Isabel and I had worked closely together for more than a year on my book, based on legal confidentiality and a process approved by the Cabinet Office. Isabel repeatedly reiterated the importance of trust throughout, and then broke that trust.”
Meanwhile Ms Oakeshott has defended her decision on the basis of helping the current public enquiry into the UK’s preparedness.
She said the text messages are “vital historical records”, and has called for “urgent answers”.
Appearing on BBC Radio 4’s Today programme on Thursday, she said: “This is about the millions of people, every one of us in this country that were adversely affected by the catastrophic decisions to lock down this country repeatedly, often on the flimsiest of evidence for political reasons,” she said.
Speaking to TalkTV, she described Mr Hancock’s statement as “a ridiculous defence”.
“For someone who’s as intelligent as Matt Hancock to issue a statement saying there is no public interest in these revelations is patently absurd,” she said. “And he knows that very well.”
Schools minister Nick Gibb defended Sir Gavin on Thursday.
“Gavin’s own wife is a primary school teacher – I’ve worked with Gavin for two years; I know he holds teachers in the highest regard,” he told radio station LBC.
“We all in Government hold teachers in the highest regard, both during the pandemic and in normal times as well.
“People say things in the heat of the moment on WhatsApp that they don’t really believe.”