Speaking at a business event in Manchester, Mr Johnson said: “Of course I’m going to go for it.”
His announcement comes as Sir Graham Brady MP, chairman of the influential 1922 Committee of the backbenchers, says the Prime Minister will be expected to announce her departure date if she loses the next Brexit vote.
Mr Brady, along with the executive of the 1922 Committee, met Mrs May earlier to discuss a departure timetable.
Mrs May has previously said she will resign once parliament backs her Brexit withdrawal deal.
However she has come under increased pressure to leave her post early after the Conservative Party lost more than 1,000 seats at the recent local government elections.
Formerly the Mayor of London, Mr Johnson acknowledged that there is not currently a vacancy in Downing Street.
Mr Johnson has repeatedly expressed leadership ambitions and was widely expected to stand in the previous race ultimately won by Mrs May. The former Foreign Secretary, who stood down his cabinet post over concerns about the government’s Brexit strategy, decided not to run in 2016 after then-ally Michael Gove MP announced a leadership bid.