Truss stood down on Thursday after losing the backing of Conservative MPs, making her the shortest serving PM in history.
In her resignation speech, she confirmed a successor would be appointed next week.
According to The Times, Johnson will announce his intention to run shortly, and considers his decision to be "a matter of national interest".
A number of Conservative MPs have backed him to take over, leading to widespread criticism from opposition figures.
Labour MP Rushanara Ali said: "[The] Tory party are a complete joke. We can’t go on like this. Time for a general election. Time for a Labour government."
Watch: How Liz Truss became the UK’s shortest serving PM
Her colleague Maria Eagle called the idea of Johnson's return "a farce".
Deputy Liberal Democrat leader Daisy Cooper said Johnson should be barred from competing in the Tory leadership contest.
“The fact that Conservative MPs are even considering putting Boris Johnson back in Number 10 shows how out of touch they really are,” she said.
“They think there’s one rule for them and another for everyone else.
“Boris Johnson was forced to resign in disgrace after countless lies, scandals and failures. He shattered public trust in the government and plunged the UK into a political crisis. He must never be allowed near Downing Street again."
Tory MP Justin Tomlinson said it was “wishful thinking” to imagine that Johnson could return to the top job.
Tomlinson, who described himself as someone who had strongly supported the former prime minister in the past, told Sky News: “I just think it’s too soon. I was there supporting him to the very end but he did lose the confidence of the majority of our colleagues.
Other candidates in the running include former chancellor and Truss leadership rival Rishi Sunak, as well as current leader of the House of Commons Penny Mordaunt.
A spokesperson for Betfair said there was a "lot of momentum for the former prime minister" whose odds had shortened to 11/2 from 27/1 in the space of an hour.
However, Sunak (11/10) remains the clear favourite, ahead of Mordaunt (9/4).
Shortly before Truss's resignation, Johnson supporter Nadine Dorries called for his return, claiming he is the only candidate with a mandate from the public.
"One person was elected by the British public with a manifesto and a mandate until January 25," said Dorries.
"If Liz Truss is no longer PM there can be no coronation of previously failed candidates. MPs must demand return of Boris Johnson – if not it has to be leadership election or a general election."
How will the next prime minister be chosen?
Sir Graham Brady, chairman of the powerful 1922 Committee of Tory backbenchers, outlined the process in a statement on Thursday afternoon.
Candidates to replace will need to gather at least 100 nominations from Conservative MPs by 2pm on Monday.
If only one candidate secures that number, they will become prime minister.
If more than one meets the threshold, Conservative Party members will choose the winner in an online ballot.
Tory party chairman Jake Berry said the board of the Conservative Party met at 4pm and, in conjunction with the 1922 Committee, had decided on the process.
“We have decided that if the party should decide to put forward two candidates there would be an expedited, binding, online vote of Conservative Party members to choose its next leader,” Berry said.
In her brief resignation speech, Truss said she will stay on as PM until a successor is chosen.