The leadership race rules mean that candidates need the backing of at least 100 conservative MPs by 2pm on Monday to be able to move to the members’ vote.
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”.
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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.”
Boris Johnson and Rishi Sunak were said to be locked in talks late into the evening as speculation grew over whether the pair would manage to strike a deal to lay the foundations for a unified Conservative government.
When it came to public support by MPs, Johnson was lagging behind his former chancellor as Sunak gained a valuable ally in Kemi Badenoch on Saturday.
Sir James Duddridge, a friend of Mr Johnson, said the former Prime Minister had gained support from 100 MPs required to keep his place in the vote.
However, Richard Holden, a supporter of Sunak’s doubted this suggestion, arguing that the equivalent number of public declarations “don’t exist” because they hadn’t been made.
As the day drew to a close, neither Johnson nor Sunak had declared their candidacy despite reports suggesting they were trying to agree on a joint ticket.
In a blow to Mr Johnson’s campaign should he decide to seek a second stint in Downing Street, International Trade Secretary and former leadership contender Ms Badenoch threw her weight behind the ex-chancellor, insisting it was not the time for “nostalgia for the cavalier elan of 2019”.
She admitted she had “on occasion” been a member of “the Boris Johnson fan club”, but she said the Tories are not “organising a popularity contest”, and stressed the party is “not a vehicle for any one individual’s personal ambitions”.
Mr Johnson returned to the UK on Saturday following rumours that he could be running for the Prime Minister role, in a move that has divided opinion among Conservative MPs including his former allies.
On Saturday morning, he arrived at Gatwick Airport with his family after cutting their holiday in the Dominican Republic short after Liz Truss’ resignation on Thursday.
Meanwhile, an ally of former home secretary Suella Braverman told the PA news agency she had been personally “heavily courted” by both Mr Johnson and Mr Sunak and was likely to decide who to back for the Tory leadership on Sunday.
Cabinet minister Penny Mordaunt was the first and only candidate to declare before Rishi Sunak declared.