The Prime Minister said the Match of the Day storm, and questions about the BBC’s impartiality, is ‘a matter for them, not the Government’.
The controversy, which has involved the broadcaster taking the sports pundit off air, began when Lineker branded the Government’s plans to tackle small boat crossings in the Channel as ‘immeasurably cruel’.
He compared the language used to launch the policy with 1930s Germany.
Commenting for the first time since the BBC stood Lineker down from Match Of The Day hosting duties, the Prime Minister said he hopes the dispute can be ‘resolved in a timely manner’.
He continued to defend his proposals to permanently ban asylum seekers who arrive in the UK on unauthorised small boats.
In a statement, Mr Sunak said: ‘As Prime Minister, I have to do what I believe is right, respecting that not everyone will always agree. That is why I have been unequivocal in my approach to stopping the boats.
‘Gary Lineker was a great footballer and is a talented presenter. I hope that the current situation between Gary Lineker and the BBC can be resolved in a timely manner, but it is rightly a matter for them, not the Government.
‘While that process is ongoing, it is important that we maintain perspective, particularly given the seriousness of the issue at hand.
‘Forty-five thousand people crossed the channel illegally last year, many of whom have been exploited or trafficked by criminal gangs, putting their lives in danger.
‘We need to break this cycle of misery once and for all and the policy we set out this week I believe aims to do just that.
‘It is not only the fair and moral thing to do, it is also the compassionate thing to do.
‘There are no easy answers to solving this problem, but I believe leadership is about taking the tough decisions to fix problems.
‘I know not everyone will always agree, but I do believe this is fair and right.’
Some of Mr Sunak’s Conservative Party colleagues have been outspoken in their criticism of Lineker, with Home Secretary Suella Braverman previously accusing him of ‘diminishing the unspeakable tragedy’ of the Holocaust.
Former culture secretary Nadine Dorries said the ex-England striker ‘does need to decide… is he a footie presenter or a candidate for the Labour Party?’
But the BBC has faced a backlash from other politicians for telling Lineker to step back from hosting the Saturday edition of the football highlights programme in the row over impartiality.
Sir Keir Starmer said the broadcaster was ‘caving in’ to Tory MPs and was ‘the opposite of impartial’.
Labour MP Jess Phillips told Times Radio: ‘If Gary Lineker had tweeted ‘stop the boats’, he would still be on air tonight.’