Britain's ruling Conservative Party is choosing a new leader after Johnson was forced to announce his resignation when ministers resigned en masse from his government amid a series of scandals and missteps over the last 12 months.
Sunak, whose resignation helped trigger Johnson's downfall, and Foreign Secretary Truss are the two remaining candidates in the contest.
The party's members will vote by postal ballot over the next few weeks with a winner announced on September 5.
"It's still early days and I'm looking forward to meeting many more of you in the coming weeks," Sunak said in a tweet ahead of campaigning among Conservative Party members later on Wednesday.
The winner of the party leadership contest will inherit a parliamentary majority and therefore become prime minister.
On Tuesday, an opinion poll showed Truss held a 34 point lead over Sunak among Conservative Party members, with 86 per cent telling pollster YouGov they had decided how they would vote.
Truss suffered the first major misstep of her campaign on Tuesday when she was forced to backtrack on one of her most striking pledges a day after announcing it following a backlash from fellow Conservatives and opposition parties.
Truss had set out plans to save billions of pounds a year in government spending in a pledge opponents said would require cutting the pay of public sector workers, including nurses and teachers, outside the wealthy southeast of England.
Sunak has struggled in the race, partly due to his role in Johnson's resignation and over his record in government.
As finance minister, Sunak implemented tax increases to pay for government support offered during the COVID-19 pandemic and to help Britons subsidise soaring energy bills.
That has drawn criticism from many activists in the historically low-tax party.
Sunak has pledged to cut taxes over time to avoid stoking inflation, but Truss has said she would act immediately to lower the burden.
Meanwhile, the Conservative Party said on Wednesday it had enhanced security around the ballot process to decide its next leader.
The Telegraph reported on Tuesday the vote had been delayed after warnings from the National Cyber Security Centre (NCSC) that hackers could change people's ballots, although the report added there was no specific threat from a hostile state.
"We have consulted with the NCSC throughout this process and have decided to enhance security around the ballot process," a party spokesman said.
"Eligible members will start receiving ballot packs this week."
The NCSC confirmed it provided advice to the Conservative Party on security considerations for the online voting.
