Suella Braverman told ITV's Peston there was currently "an overwhelming sense of despair among Conservative MPs," adding: "The time has come for the Prime Minister to step down".
Ms Braverman said it was "untenable" for Mr Johnson to continue as leader - and signalled her own intention to run in a future contest.
The Cabinet minister, who has previously been a staunch supporter of the PM, said he had handled matters “appallingly” in recent days.
"The balance has tipped now in favour of saying that the Prime Minister – it pains me to say it – but it’s time to go."
Also speaking to ITV's Peston Matt Hancock, a former health secretary in Mr Johnson's government, said the PM should "of course" stand down.
"Sadly he's lost the support of his cabinet, many, many of his ministers and the parliamentary party... it's come to the point where he needs to resign," Mr Hancock said.
Meanwhile Ms Braverman, MP for Fareham, also told ITV's Peston she would put her name into the ring in any future leadership contest.
"If there is a leadership contest, I will put my name into the ring," Ms Braverman said.
"I love this country, my parents came here with absolutely nothing and it was Britain that gave them hope, security and opportunity. This country has afforded me incredible opportunities in education and in my career. I owe a debt of gratitude to this country and to serve as PM would be the greatest honour, so yes, I will try."
Mr Hancock said he would not be standing in a future leadership election, but said: "I do care deeply about what happens next" adding he'd be "happy" to serve in government.
The PM has faced more than 40 resignations from his government as of Wednesday night, but supporters told ITV News Mr Johnson was still in "quite a buoyant mood" and was "up for a fight".
Ms Braverman said she would not be resigning from her role, despite calling for the PM to go, as she said she has a "duty" to continue.
"I don’t want to resign because I have that duty and we need an Attorney in government," she said.
"My first duty is to the country... we're in a crisis and I have statutory, legal and constitutional duties," Ms Braverman told ITV's Peston. "We need an attorney in government," she said.
Quizzed on there being a "good chance" the PM would now ask her to resign, Ms Braverman said: "That is his choice and I will do whatever the PM asks me to do".
A reshuffle was underway in Number 10 on Wednesday night as the PM looked to fill the vacancies in his cabinet left by resignations.