Mr Sunak tweeted: “The United Kingdom is a great country but we face a profound economic crisis. That’s why I am standing to be leader of the Conservative Party and your next prime minister.
“I want to fix our economy, unite our party and deliver for our country.”
Mr Johnson’s supporters have downplayed the fact he is lagging behind his former chancellor in public support from MPs, saying he is “clearly” running, although he has not officially declared this.
Mr Sunak, becoming the second candidate to enter the race after Penny Mordaunt, said he wants to “fix our economy, unite our party and deliver for our country”.
He said in a statement: “I served as your chancellor, helping to steer our economy through the toughest of times.
“The challenges we face now are even greater.
“But the opportunities, if we make the right choice, are phenomenal.
“I have the track record of delivery, a clear plan to fix the biggest problems we face and I will deliver on the promise of the 2019 manifesto.
“There will be integrity, professionalism and accountability at every level of the government I lead and I will work day in and day out to get the job done.”
Mr Sunak on Sunday gained the valuable backing of Northern Ireland minister Steve Baker, an influential figure on the Tory right as the former head of backbench Brexiteers.
Mr Baker was scathing in his assessment of what a comeback by Mr Johnson would mean.
He told Sky’s Sophy Ridge On Sunday: “Boris would be a guaranteed disaster.
“There’s going to be a vote before the House of Commons on this issue of privileges, whether he will deliberately misled the house.
“In that vote it’s guaranteed there’ll be a large number of Conservatives who will refuse, as they see it, to lay down their integrity to save him, and at that moment his premiership will collapse.”
Mr Johnson is to face an inquiry into whether he lied to the Commons over the “partygate” scandal, for which he was fined by police.
If found guilty by the Commons Privileges Committee, he could face recall proceedings that would leave him battling for his seat in the Commons if he receives a suspension of 10 days or more.
“I’m not willing to lay down my integrity for Boris Johnson,” Mr Baker said.
He added that “this isn’t the time for Boris and his style”, noting that Mr Johnson does not have the capacity to comply with “tedious rules”.
Jacob Rees-Mogg confirmed Mr Johnson intends to run, told the BBC’s Sunday With Laura Kuenssberg programme: “I have been speaking to Boris Johnson, and clearly he’s going to stand, there’s a great deal of support for him.”
It came as Nadhim Zahawi, the Chancellor of the Duchy of Lancaster, became Mr Johnson’s latest Cabinet backer as he said he “got the big calls right” and argued “Britain needs him back”.