The pair, who revealed their plans to run in the Sunday Mail, are the first to declare their candidacy.
Justice Secretary Keith Brown, Neil Gray, and Mairi McAllan have ruled themselves out.
The winner of the race is due to be announced on 27 March.
Finance Secretary Kate Forbes is expected to announce her candidacy as early as Monday, according to BBC political editor Glenn Campbell.
Angus Robertson, the constitution secretary, who is also tipped for the job, is said to be undecided.
Announcing his leadership bid in the Scottish newspaper, Mr Yousaf says he had been through a "rollercoaster of emotions" since the longstanding first minister and SNP leader Ms Sturgeon announced her shock resignation on Wednesday, after eight years.
He says: "You've got to put yourself forward if you think you're the best person for the job. And I do. This is the top job in the country, and it needs somebody who has experience."
Mr Yousaf, who first became an MSP in 2011, has been a prominent figure on the SNP frontbenches in every Scottish administration since and has been considered a strong leadership candidate for many years.
But the 37-year-old's time as justice minister saw him bogged down in controversy surrounding the Hate Crime and Public Order (Scotland) Bill - which faced backlash over its impact on freedom of expression.
He has also been criticised for his running of the NHS in Scotland, which experienced the hardest winter in its history in recent months.
Ms Sturgeon has faced repeated calls from opposition parties to sack Mr Yousaf, with waiting times at record highs and doctors warning that the country's hospitals are not safe for patients.
On Friday, Mr Yousaf pointed to the record pay offer he made to NHS staff, which he said was likely to avoid strike action for the next financial year.
Ivan McKee, the minister for business, told BBC Scotland that he hoped the "hugely talented" Ms Forbes would put her name forward.
"In my mind she is head and shoulders ahead of all the other candidates," he said.
He said there was a broad range of support for Ms Forbes in the party and beyond.
Mr McKee added that he did not think her beliefs as a member of the Free Church of Scotland should prevent her from taking on the role.
"We're in a bad place if we are deciding that people can't stand for political office based on whatever faith they may hold," he said, adding that she was the only candidate whose religion or young family had been raised as an issue.
Nominations for the leadership contest will close at noon on Friday.
A party conference scheduled for 19 March to discuss the SNP's strategy for winning Scottish independence has been postponed.