Gary Lineker is facing demands to be pulled off air during the next general election over concerns he will not abide by the BBC's impartiality rules.
The Match of the Day host sparked outrage last week after he signed a letter calling for the Government to scrap its Rwanda scheme.
He joined 30 others in signing the letter which demanded political leaders come up with a "fair new plan for refugees".
Incoming BBC chairman Samir Shah said that Lineker appeared to have broken the broadcaster's guidelines, with pressure now mounting on the corporation to take action.
Gary Lineker caused outrage last week after he signed a letter calling for the Government to scrap its Rwanda scheme
PA
Conservative MPs say that the 63-year-old must not be allowed to present on TV during the general election after having repeatedly proven he is willing to wade into political matters.
Stoke-on-Trent North MP Jonathan Gullis told the Mail on Sunday: "The incoming BBC chairman thinks that Gary Lineker has breached the social media guidelines.
"The BBC must be willing to take him off air for the next General Election."
Calling the presenter's position "untenable", Ipswich MP Tom Hunt agreed, stating: "Gary Lineker should have been shown the red card by the BBC months ago.
"His position as a publicly funded presenter has clearly become untenable and he should have been sacked long ago.
"The feebleness of the BBC leadership over this issue will further erode support for the licence fee."
Meanwhile, Rother Valley representative Alexander Stafford said: "At the very least, Gary Lineker should undertake not to tweet his opinions for the duration of the General Election campaign.
"Failing that, the BBC should itself impose a strict social media ban on the presenter for that period."
The BBC faced an internal revolt earlier this year when it suspended Lineker from Match of the Day for a tweet comparing the Government's migration plan to Nazi Germany.
The broadcaster reinstated the former England footballer after staff walked out en masse in protest at its decision.
New social media guidelines were drawn up as a result.
Following Lineker's latest foray into politics, the BBC said that it does not comment on "individuals or indeed individual tweets".
However, a spokesman added: "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."