The controversial right-winger – who quit as deputy Tory chairman last week over the bill – said he would be willing to return if wanted by the PM.
And No 10 even hinted that Mr Sunak is open to bringing back Mr Anderson in the senior party role – despite all the turmoil of the past week.
The MP for Ashfield resigned after supporting a rebel amendment aimed at toughening up the legislation, and then abstained on the crunch vote in the Commons.
Expressing his regret over the rebellion, Mr Anderson told The Telegraph on Wednesday that should have been “brave” and sided with Mr Sunak after all.
Asked if he would return to his old role if approached by Mr Sunak, Mr Anderson said: “Yeah, of course I would.”
He added: “I did wrong. Well, I say I did wrong, I acted on a point of principle. I had to resign. I had no choice. I bear no malice or anything, it’s just I know the rules.”
Responding, No 10 said the prime minister still has “a lot of time” for Mr Anderson despite his resignation over the Rwanda bill.
Asked whether Mr Sunak would consider giving Mr Anderson his job back, the PM’s press secretary said it was "the PM’s prerogative to choose and when" who gets which jobs.
“I think we can say that we have a lot of time for Lee. He made it clear that he had concerns but actually he really supports getting this deterrent up and running so it’s good to see him talking about that today,” she said.
Mr Anderson said last week that he could not commit to rebelling in full against the Rwanda bill and join the 11 Tory rebels because Labour MPs were “giggling” at him.
The right-winger said “the Labour lot were giggling and laughing and taking the mick and I couldn’t do it”.
The Liberal Democrats mocked Mr Anderson’s apparent U-turn, and his sensitivity in the face of Labour jibes during the voting process.
“It seems Lee Anderson must have invested in a pair of flip flops recently,” said Lib Dem home affairs spokesperson, Alistair Carmichael.
“First cowering from Labour jeers in the voting lobbies and now begging for his old job back,” he added. “For someone who wants to be seen as a conviction politician, these past weeks Anderson has shown very little.”
Meanwhile, Mr Anderson said it would be “absolute madness” to change the leader of the Conservative Party ahead of the next general election.
Dismissing Sir Simon Clarke’s call for Mr Sunak to be replace, Mr Anderson said: “That means four different prime ministers in five years. That’s absolute madness. They’d be no trust in our party whatsoever, if it was to swap leaders again.”