Oct 22: The ex-chancellor has 93 endorsements from Tory MPs so far, though a campaign source said he had already reached 100.
Former leader Boris Johnson, second with 44 backers, is poised to enter the contest to succeed Liz Truss and is flying back from a Caribbean holiday.
Penny Mordaunt was the first to declare, counting 21 supporters so far.
Trade Minister Sir James Duddridge, who is backing Mr Johnson, told BBC News: "He is coming home and is up for it."
Neither Mr Sunak nor Mr Johnson have officially launched their campaigns, but this has not stopped backers declaring their support.
Mr Johnson is on a flight back from a holiday in the Dominican Republic, and is due to arrive in London on Saturday morning.
A BBC reporter on the flight says some passengers who tried to take selfies were stopped from doing so by his sizeable security team.
MPs will be waiting to see if he makes a statement when he lands.
Sources from Mr Sunak's campaign told BBC News he had already reached the 100 nominations needed to make it onto the ballot paper.
Mr Sunak secured support from a number of senior colleagues, including former chancellor and health secretary Sajid Javid, Security Minister Tom Tugendhat and former health secretary Matt Hancock.
"It is abundantly clear that Rishi Sunak has what it takes to match the challenges we face - he is the right person to lead our party and take the country forward," Mr Javid said.
Another backer, Tobias Ellwood, said he was the 100th MP to endorse Mr Sunak, saying: "Time for centrist, stable, fiscally responsible government offering credible domestic & international leadership."
Mr Johnson's supporters said the former prime minister had "momentum and support". Mr Duddridge said: "He is the only election winner we have that has a proven track record in London, on Brexit, and in gaining the mandate we have now."
'I'm up for it'
Sir James also read out what he said was a message from Mr Johnson to the PA Media news agency that appeared to confirm his bid: "I'm flying back, Dudders. We are going to do this. I'm up for it."
The former Tory party leader is said to have contacted MPs from his holiday, the i newspaper reported, promising colleagues that if elected he would "adopt a more inclusive style of governing and bring more discipline to the Downing Street operation".
International Trade Secretary Kemi Badenoch and former Home Secretary Suella Braverman are said to be considering backing Mr Johnson, in a move that could unify the Right of the Tory party, the Daily Telegraph reported.
Tory MP Brendan Clarke-Smith told BBC Newsnight that Mr Johnson has not officially declared his campaign because "he's probably waiting to see whether he felt he'd got the support... I'd say he's certainly up for it".
Meanwhile, Rishi Sunak backer Tory MP Craig Williams said he will likely respond to calls to run for leader "in the coming hours or days... there are clearly over a hundred colleagues publicly declaring parliamentary support."
Declaring her run earlier, Ms Mordaunt, the current leader of the House of Commons, said she had been encouraged by colleagues to run for prime minister - and pledged to "unite our country, deliver our pledges and win the next [general election]".
She later told the Daily Telegraph: "We owe it to the country to have a detailed plan of how we will deliver. It is not enough to tell people we understand their issues. Our success should be measured in whether people really feel that we can support them.
"I will harness the talents of all sides of the Conservative Party, and all the talents of the country to deliver this."
Penny Mordaunt backer Conservative MP Bob Seely said "I think we owe the country a collective responsibility to apologise" and said he believes Ms Mordaunt has the best chance of providing "unity and leadership" within the party.
Among those to have ruled themselves out of the race are Defence Secretary Ben Wallace - who has indicated he is "leaning" towards supporting Mr Johnson - and current Chancellor Jeremy Hunt.
Chancellor Jeremy Hunt has ruled himself out of the race, along with Mr Tugendhat and former cabinet minister Michael Gove.
Contenders have until 14:00 BST on Monday to find 100 backers. If three reach the threshold, Conservative MPs will knock out one contender in a ballot on the same day.
MPs will hold an "indicative" ballot of the final two, with the winner then decided in an online vote of party members, to finish next Friday.
JULY 7: Boris Johnson is to stand down as Conservative Party leader after losing the support of his ministers and MPs.
A Conservative leadership contest will take place this summer and a new prime minister will be in place in time for the party conference in October.
In the meantime, Mr Johnson will continue as prime minister.
He had vowed to "keep going" following a wave of resignations from the government over his leadership but has now decided to step down.
But was urged to stand down by senior members of his cabinet, including newly-appointed chancellor Nadhim Zahawi.
In his resignation letter, Mr Zahawi, who was given the job less than 48 hours ago, said he had "made clear to the prime minister" that he should "leave with dignity".
Mr Johnson is expected to give a resignation statement outside No 10 Downing Street later.
With inputs from BBC
Prime Minister Boris Johnson is facing British lawmakers on Tuesday for the first time since he was fined by police for attending a birthday party in his office that broke coronavirus lockdown rules.
British Prime Minister Boris Johnson faced a high-stakes showdown Wednesday in Parliament that could help defuse a crisis over claims that he and his staff broke coronavirus lockdown.
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