Lineker had recently criticised the UK government's new asylum policy, comparing the language of the bill to 'that used by Germany in the 30s.'
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In the aftermath, the former England striker told reporters he was not worried about the potential of him being suspended.
However, amid the impartiality row, a spokesperson for the BBC said: "The BBC has been in extensive discussions with Gary and his team in recent days.
"We have said that we consider his recent social media activity to be a breach of our guidelines.
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"The BBC has decided that he will step back from presenting Match of the Day until we've got an agreed and clear position on his use of social media.
"When it comes to leading our football and sports coverage, Gary is second to none.
"We have never said that Gary should be an opinion free zone, or that he can't have a view on issues that matter to him, but we have said that he should keep well away from taking sides on party political issues or political controversies."
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Lineker's co-star Ian Wright was quick to show his support on social media after the announcement, saying he will not feature on this Saturday's show.
He tweeted: "Everybody knows what Match of the Day means to me, but I've told the BBC I won't be doing it tomorrow. Solidarity."
Laura Woods was among those to praise Wright for his decision, as the talkSPORT host and Arsenal fan replied with a series of clapping hands emojis.
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It remains unclear as to who will fill in Lineker on this Saturday's edition of Match of the Day.
He is a freelance broadcaster rather than a permanent employee at the BBC and because he is not in charge of news or political content, he does not need to stick with the same impartiality rules.
Former Manchester United star Gary Neville commented on the decision: "When you take on the Tories and the system! Awful people who we need gone. Off for a [wine emoji]."
TalkTV host Piers Morgan also had his say: "WTF? How pathetically spineless.. I now demand the BBC suspend every presenter who has made public comment about news or current affairs - starting with Sir David Attenborough and Lord Sugar.'"
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In the days after his comments were met with criticism, Lineker said on Twitter: "I have never known such love and support in my life than I'm getting this morning [England World Cup goals aside, possibly].
"I want to thank each and every one of you. It means a lot. I'll continue to try and speak up for those poor souls that have no voice. Cheers all."
Lineker has been presenting Match of the Day since the late 1990s, the longest tenure out of all of their presenters.
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He is therefore a regular feature on TV screens on Saturday nights when the BBC broadcast highlights of the Premier League's action.