Mr Johnson was trying to persuade his former chancellor that he should return to No 10 and that Mr Sunak will secure a top job if he is successful, the Mail on Sunday reported.
The former prime minister arrived at Gatwick Airport on Saturday morning with his family after breaking off a holiday in the Dominican Republic in the wake of Ms Truss’s dramatic resignation on Thursday.
However prominent Tory donors were on Saturday night pleading with Mr Johnson - whose three years in power were mired in scandal - not to stand against Mr Sunak, as reported by The Sunday Times.
Sir Mick Davis, a former treasurer of the party and a donor, told the newspaper that the idea of Mr Johnson forming a stable government was “delusional”.
Mr Johnson suffered another blow on Saturday evening when Kemi Badenoch, who is popular on the right of the party, threw her support behind Mr Sunak.
Writing in The Sunday Times, Ms Badenoch said her party must remind people that “Conservatives care about the country, not ourselves”.
“Everything we do must be for the greater good and must be seen to be for the greater good, rather than just winning elections,” she said.
The International Trade Secretary suggested Mr Sunak would bring a “disciplined approach” to government, citing his “fiscal conservativism” and stressing that “right now, being able to say no is what we need”.
It comes as Sir James Duddridge, a Johnson ally, claimed the ex-PM had the backing of the 100 MPs required to make it on to Monday’s ballot.
But Sunak supporter Richard Holden cast doubt on this suggestion, arguing the equivalent public declarations had not yet manifested “because they don’t exist”.
Sir Robert Syms, another Sunak backer, wrote on Twitter: “If Boris has 100 in the bag why is his campaign putting out pics of him begging for votes?”
Despite being the only candidate to declare so far, Cabinet minister Penny Mordaunt is lagging behind on public support from MPs, with just 21 to Mr Johnson’s 43 and Mr Sunak’s 110, according to a PA.
Setting out her plan to “unite the party and the country” in The Express, she warned the Tories have “let ourselves become distracted by internal disputes”.
Ms Mordaunt used her pitch to stress the need to “make Brexit work”, “focus on the potential of all our citizens” and “defend our Union and its territorial integrity”, pledging her support for reforming the controversial Northern Ireland Protocol.
She insisted she is not seeking the top job for an “easy ride”, and vowed to build a government which “draws from all our best talent”.
“As prime minister, I will create that team of Conservatives, we will get real, and we will get to work,” she added.
Tory MPs will vote on Monday, and two candidates will be put forward to the Tory membership unless one pulls out, with a result being announced on Friday.
Additional reporting by PA