But widespread uncertainty remained after British media reported that Sunak had held late-night talks with Johnson on Saturday, and speculation mounted that the pair could strike a deal to unite the fractured governing party after it was left reeling from Truss' rapid downfall.
The Conservative Party has ordered a contest that aims to finalise nominations on Monday and install a new prime minister - its third this year - within a week.
Sunak, 42, confirmed on Sunday he was running in the leadership race. He has the backing of at least 124 Conservative lawmakers, according to unofficial tallies by the BBC and Sky News. That is well ahead of the 100 nominations required to qualify.
"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," Sunak said in a statement.
Mordaunt garnered about 24 lawmakers' public support, while Johnson, who has not declared if he is running, has about 50 so far.
Lawmaker Jacob Rees-Mogg told the BBC Sunday he spoke to Johnson and "clearly he's going to stand".
The return of Johnson, 58, who was forced out of office just weeks ago by a string of ethics scandals, has divided the Conservatives and thrown unpredictability into the race.
Supporters say he is a vote winner and has enough support from lawmakers, but many critics warn that another Johnson government would be catastrophic for the party and the country.
Northern Ireland minister Steve Baker, a former backer of Johnson and an influential politician within the Conservative Party, warned a Johnson comeback would be a "guaranteed disaster" because he still faces an investigation into whether he lied to parliament while in office that could lead to his suspension as a lawmaker.
"This isn't the time for Boris and his style," Baker told Sky News on Sunday. "What we can't do is have him as prime minister in circumstances where he's bound to implode, taking down the whole government within and we just can't do that again."
Truss quit on Thursday after a turbulent 45 days, conceding that she could not deliver on her tax-cutting economic package, which she was forced to abandon after it sparked fury within her party and weeks of turmoil in financial markets.
Dozens among Britain's 357 Conservative lawmakers have not yet publicly declared whom they are backing to replace Truss.
Mordaunt and Johnson - if he confirms he is running - have until Monday afternoon to garner 100 nominations.
If all three meet that threshold, lawmakers will vote to knock out one and then hold an indicative vote on the final two.
The party's 172,000 members will then get to decide between the two finalists in an online vote. The new leader is due to be selected by Friday.
