Sunak, whose exit from the Treasury last week was one of the first in a cascade of resignations that forced Johnson to quit, has been the frontrunner among Conservative lawmakers despite being blamed by some for his role in Johnson's downfall.
He faces competition from Penny Mordaunt, a junior trade minister, and foreign minister Liz Truss, who launched her official campaign earlier arguing she was the only contender with the experience to take the "tough decisions" needed.
There are now five hopefuls, after Brexit supporter Attorney General Suella Braverman was knocked out of the race on Thursday for failing to reach the 30-vote threshold in the second round.
Whoever gets the job will take on rocketing inflation and low economic growth, as well as the public's lack of confidence in politics after Johnson's scandal-ridden time in power.
After maintaining his lead Sunak thanked his supporters, and said on Twitter: "I am prepared to give everything I have in service to our nation. Together we can restore trust, rebuild our economy and reunite the country."
Earlier, he took to the airwaves to say his first economic priority would be tackling high inflation, not the tax cuts pledged by his rivals.
Truss, the last hopeful to launch her official campaign, is the latest to also promise to restore trust in politics, trying to distance herself from the increasingly personal and bitter briefings and counter-briefings from the opposing camps.
"I will campaign as a Conservative and I will govern as a Conservative. I can lead, I can make tough decisions and I can get things done. I am ready to be Prime Minister from day one," Truss said.
Asked why she did not resign when support disappeared for Johnson prompting last week's resignation, she said: "I'm a loyal person. I'm loyal to Boris Johnson."
Still a way to go
Those remaining - who also include former equalities minister Kemi Badenoch and Tom Tugendhat, chair of parliament's Foreign Affairs Committee - will face a third round of voting by Conservative members of parliament on Monday.
If all the candidates reach the 30-vote barrier, the one with the lowest number of votes will be dropped from the contest. Subsequent ballots will eliminate the candidate with the fewest votes each time, to whittle the field down to a final two by July 21.
The new leader will then be chosen from those two by the country's 200,000 Conservative Party members, and will be announced on September 5.
Sunak might be the most popular contender with his colleagues, but a YouGov poll of nearly 900 party members found Mordaunt was the favourite, beating any of the others in a run-off. She had a huge lead over Sunak, who fared badly against almost all his rivals, and she is now the bookmakers' favourite.
Britain's economy is beset with spiralling inflation, high debt, and low growth that have left people grappling with the tightest squeeze on their finances in decades. Fuel prices have soared in an energy crunch exacerbated by the war in Ukraine.
The leadership campaign is offering a glimpse of the difficulties the new Prime Minister might face, and of rifts within the party. Mordaunt fended off criticism from former Brexit negotiator David Frost that she was insufficiently tough towards the European Union.
Tugendhat, who ranked fifth in the second round of voting, said, "It is fair that we (candidates) receive criticism ... I am asking for the trust of the British people for the highest job in the country," he told reporters.
"When you are making a big ask you should expert scrutiny and expect to present yourself to answer to questions."
The contenders
KEMI BADENOCH
Badenoch was elected to parliament for the first time in 2017. She has held junior ministerial jobs, including most recently minister for equalities, but has never served in Cabinet.
A former Conservative member of the London Assembly, she has also served as vice-chair of the Conservative Party. The 42-year-old supported leaving the European Union in the 2016 referendum.
She says she believes in limited government "focused on doing some things well, not lots of things badly". She has said she wants tax cuts, but only if government spending also comes down, and has criticised net-zero emissions targets and restrictions on free speech.
PENNY MORDAUNT
The former defence secretary was sacked by Johnson after she endorsed his rival, Jeremy Hunt, during the 2019 leadership contest.
Mordaunt (49) was a passionate supporter of leaving the EU and has said she will aim to deliver the benefits of Brexit and help Britain recover from economic shocks such as the pandemic.
Currently a junior trade minister, Mordaunt called the Covid lockdown-breaking parties in government "shameful" and has said that if she is prime minister, leadership has to change to be less about the leader.
RISHI SUNAK
In his campaign launch video, Sunak promised to confront the difficult economic backdrop with "honesty, seriousness and determination", rather than piling the burden onto future generations. Read full story
The 42-year-old became finance minister in early 2020 and was praised for a Covid economic rescue package, including a costly jobs retention programme that averted mass unemployment.
But he later faced criticism for not giving enough cost-of-living support to households. Revelations this year about his wealthy wife's non-domiciled tax status, and a fine he received for breaking Covid lockdown rules, have damaged his standing.
His tax-and-spend budget last year put Britain on course for its biggest tax burden since the 1950s, undermining his claims to favour lower taxes. Sunak voted to leave the EU in the 2016 referendum.
LIZ TRUSS
The foreign secretary has been the darling of the Conservative Party's grassroots and has regularly topped polls of party members carried out by the website Conservative Home.
Truss has a carefully cultivated public image and was photographed in a tank last year, echoing a famous 1986 photo of Thatcher.
She spent the first two years of Johnson's premiership as international trade secretary and is now in charge of dealing with the EU over post-Brexit trade rules for Northern Ireland, where she has taken an increasingly tough line in negotiations.
Truss (46) initially campaigned against Brexit but after the 2016 referendum said she had changed her mind.
TOM TUGENDHAT
The chair of parliament's foreign affairs committee, and a former soldier who fought in Iraq and Afghanistan. However, he is relatively untested because he has never served in Cabinet.
Tugendhat, 49, has been a regular critic of Johnson and would offer his party a clean break with previous governments.
He says he is a low-tax Conservative who did not support the rise in National Insurance, and has said fuel tax is "crippling" for many people. He voted to remain in the EU.
