The pair, who revealed their plans to run in the Sunday Mail, are the first to declare their candidacy.
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.
For her part, Ms Regan tells the Sunday Mail the SNP “need to bring back unity, draw a line under certain things and move past them”, adding her belief that she is “the person to do that”.
This statement is a reference to her views over the controversial the Gender Recognition Reform (Scotland) Bill, the SNP’s proposed reforms to how people can change their legal gender in Scotland – which was subsequently blocked by the Westminster government.
Before it was approved by a majority in Holyrood, the SNP’s former community safety minister had quit in protest over it and became an outspoken critic of the legislation.
Writing in the Sunday Mail, Ms Regan says: “The electorate expect the Scottish government to focus on things that are important to them.
“That means the NHS, which is still struggling to get back on its feet after the pandemic. People expect a first minister to concentrate on boosting the economy, creating jobs and helping them deal with the cost-of-living crisis.”
In a later post on Twitter she outlined a plan to call for independence convention to be held to “create a new vision of an independent Scotland”.
And she backed proposals to use either a Westminster or Holyrood election as a de facto referendum on Scottish independence.
After Ms Sturgeon’s announcement, Ms Regan, 48, called for SNP members who left the party in the past year to be given a vote in the leadership race.
But this proposal was described as “preposterous” by Deputy First Minister John Swinney – who has already ruled himself out of the contest.
After days of speculation and background briefings, two candidates to succeed Nicola Sturgeon have now declared.
Humza Yousaf is seen as part of a new generation of SNP talent, albeit with a decade of ministerial experience.
Many view him as something of a continuity candidate – one who in top office would advance much of the government’s existing agenda.
If Humza Yousaf is a party establishment figure, Ash Regan is the opposite.
After quitting her ministerial post over the Scottish government’s gender reforms, her leadership pledge involves ditching the policy. A move that would surely mean the end of the SNP’s cooperation deal with the Scottish Greens, such is their commitment to the Gender Recognition Reform Act.
Their contrasting approaches look set to make for an interesting contest. And that’s before others join the race. Kate Forbes is expected to announce her candidacy imminently.
Meanwhile Constitution Secretary Angus Robertson, with a wealth of leadership experience at Westminster, is said to be undecided.
The SNP’s Westminster leader Stephen Flynn and high-profile MP Joanna Cherry have also said they will not run for the leadership.
Kate Forbes, the finance secretary, who is currently on maternity leave, has been seen as one of the favourites to replace Ms Sturgeon.
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.
