Matt Hancock suggested to an aide that they "frighten the pants off everyone", messages published by the Sunday Telegraph show.
It appears the former health secretary discussed when to reveal the existence of the Kent variant of Covid, to ensure people comply with lockdown rules.
In another exchange, the head of the Civil Service, Simon Case, suggested the "fear/guilt factor" was vital to the government's messaging.
The BBC has not verified the messages.
More than 100,000 WhatsApp messages were leaked to the Telegraph by the journalist Isabel Oakeshott, who has been a vocal critic of lockdowns.
The former health secretary has repeatedly criticised the leaks, referring to the published messages as a "partial, biased account to suit an anti-lockdown agenda."
In an exchange between Mr Hancock and an aide from 13th December 2020 - five days before the government scrapped plans to relax rules for many over Christmas - the former health secretary discusses when to "deploy" the announcement of the new variant.
They are talking about the possibility of the London Mayor Sadiq Khan resisting a possible lockdown for London.
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The Department of Health adviser suggests: "Rather than doing too much forward signalling, we can roll pitch with the new strain."
Mr Hancock says: "We frighten the pants of everyone with the new strain."
The adviser responds: "Yep, that's what will get proper behaviour change."
The minister then asks: "When do we deploy the new variant."
Mr Hancock announced the new variant the following day.
In a separate WhatsApp conversation from January 2021, when lockdown measures were in place, Mr Hancock is seen discussing possible changes with Simon Case.
Mr Case warns against making small changes to the rules as looking "ridiculous". He talks about "ramping up messaging" adding the "fear/guilt factor" was "vital".
In a statement responding to the leaks, Matt Hancock said: "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."
Chris Heaton-Harris, the government's Northern Ireland secretary, told BBC's Sunday with Laura Kuenssberg the messages give "almost a view into the psyche of Matt Hancock rather than into the actual decision-making".
"I think viewers would expect that politicians being human beings would express things in a human way."
The former Conservative MP Ann Widdecombe, told the BBC's Stephen Nolan on Radio 5 Live that she had been "just as much against lockdown as Isabel Oakeshott" but found the leaks "profoundly unhelpful".
Labour's Jonathan Ashworth - who was shadow health secretary during the pandemic - said there was always "two sides to a story" but many people will be "deeply troubled" by the messages.
"I also think Rishi Sunak needs to get a grip of this situation and insist that all ministers hand over everything, that no WhatsApps are deleted."
In other newly-released leaked messages, Boris Johnson spoke of the need to get "absolutely militant" on social distancing in Covid hotspots, saying there had been a "general collapse" in rule following.
In a WhatsApp conversation with Simon Case from July 2020, he wrote: "We need to tell people that if they want to save the economy and protect the NHS then they need to follow the rules.
"And we may need to tighten the rules. You can now have 6 people from different households indoors. Do people really understand that and are they observing it?" he asked.
The exchange came one month after the PM broke the rules himself.
Mr Johnson, his wife Carrie, and the then-chancellor, Rishi Sunak, all received one fine each for attending a birthday party thrown in the ex-PM's honour in June 2020.
A spokesman for Boris Johnson said it was not appropriate to comment on these leaks, and added that the public inquiry provided the right process for these issues to be examined.
