The TV host, 63, found himself at the centre of yet another Twitter storm after urging ministers to scrap their Rwanda scheme and come up with a “fair new plan for refugees”.

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In a letter which is campaigning for change, Together With Refugees branded Britain’s refugee system “ever-more uncaring, chaotic and costly”, and said asylum policies are not working.

And Lineker landed himself in hot water after co-signing a letter with other celebs, calling for the scheme to be thrown out.

Now, BBC bosses are “talking to Lineker” about his posts, the Times reports.

The former England star turned pundit said about the letter: “We need a new system that reflects the will of the British people who have opened their homes, donated and volunteered in their local communities.

“That’s why I’m backing this new campaign, because fair really can begin here.”

The Beeb are now said to be discussing whether their updated social media guidelines, which came into play in September, are tight enough after Lineker is accused of ridiculing cabinet minister Grant Shapps and Conservative backbenchers.

Talks between bosses and Lineker about his tweets are reportedly set to take place.

A BBC spokesman said: “We aren’t going to comment on individuals or indeed individual tweets.

“While the guidance does allow people to talk about issues that matter to them, it is also clear that individuals should be civil and not call into question anyone’s character.

“We discuss issues that arise with presenters as necessary.”

Lineker previously sparked an impartiality row in March after saying the Government rhetoric on immigration was similar to that of 1930s’ Germany.

But after a review, BBC bosses decided to allow its top stars to share their own views on social media and only ruled they should stop short of political campaigning.

Grant Shapps said on Monday morning that Lineker should stick to presenting football.

When asked if the Match of the Day host should continue expressing political views, Defence Secretary Mr Shapps told Times Radio: “He’s been through all of this before.

“The BBC have told him he shouldn’t do this type of thing but still it continues.

“The point I would make to Mr Lineker is: what is right or moral about having people trafficked dangerously across the English Channel, losing their lives at sea, illegally entering the country?

“That is not a civilised, morally correct thing to do.”

He added: “I just fundamentally disagree with him. What happens to him next is up to the BBC.

“As far as I see it, they have issued previous warnings to him, so it’ll be interesting to see what they do and say at this point.

“I know millions of people watch him for his football commentary and TV presenting, I would have thought it’s better to stick with that.”

Tory deputy chairman Lee Anderson told the Daily Mail: “For once in his life, Gary’s absolutely right – we do need a system that reflects the will of the British people. What the people want is to stop the boats and to tell overpaid crisp salesmen to put a sock in it.

“Alongside cracking down on illegal migration, we need another robust system which keeps Lineker as far away from the public as possible, to give us all a rest from his Left-wing, out-of-touch nonsense.”

The Beeb row comes as the Prime Minister is trying to convince his own backbenchers to support his Rwanda Bill, with both the right of the party and the One Nation faction set to announce whether they support it later on today.

The Bill seeks to declare Rwanda a safe country and also empowers ministers to ignore parts of the Human Rights Act.

Mr Shapps told Sky News Home Office modelling suggests that of the current cases which are challenged successfully, “99.5% of them would not be challenged once this is in place”.