Sunak got 115 votes in the third ballot of Conservative lawmakers on Monday, ahead of former defence minister Penny Mordaunt on 82 and Foreign Secretary Liz Truss on 71.
The current prime minister appeared most keen on Liz Truss, the foreign secretary, endorsed by his fiercest cabinet allies.
Fellow Indian-origin candidate, Attorney General Suella Braverman, features last on the latest tally.
Former finance minister Rishi Sunak is the bookmakers' favourite, and among those he will be taking on are his successor Nadhim Zahawi and foreign minister Liz Truss.
So far 11 candidates have thrown their hat in the ring to succeed Johnson as leader of the ruling Conservative Party and prime minister after he quit.
There are a total of eight candidates now vying for the position of PM, and at least two more are expected to join the race.
The former UK Chancellor, who stepped down earlier this week triggering a deluge of government exits, is seen as a credible contender and a frontrunner in some party quarters.
"The process of choosing that new leader should begin now," Johnson said at the door of Number 10 Downing Street.
Chancellor Rishi Sunak and UK Health Secretary Sajid Javid have expressed loss of confidence in Johnson's leadership.
Meanwhile, British Prime Minister Boris Johnson made a surprise visit to Kyiv on Friday and offered a stepped-up training for Ukrainian forces.