Gary Lineker has said he will try to keep speaking up for people with "no voice", after criticism over his tweets on the government's asylum policy.
The Match of the Day host had said the language setting out the plan was "not dissimilar to that used by Germany in the 30s".
The remarks drew criticism, including from Home Secretary Suella Braverman.
The BBC, which has impartiality guidelines, said it was having a "frank conversation" with Lineker.
On Tuesday, the government outlined its plans to ban people arriving via an illegal route from claiming asylum in the UK.
The measure is part of attempts to address a rise in the number of people crossing the Channel in small boats.
The presenter described it on Twitter as an "immeasurably cruel policy directed at the most vulnerable people in language not dissimilar to that used by Germany in the 30s".
Ms Braverman said on Wednesday morning his comments were disappointing.
She told BBC One's Breakfast: "I'm disappointed, obviously. I think it's unhelpful to compare our measures, which are lawful, proportionate and - indeed - compassionate, to 1930s Germany. I also think that we are on the side of the British people here."
Later on Wednesday, Lineker tweeted: "Great to see the freedom of speech champions out in force this morning demanding silence from those with whom they disagree."
He followed up shortly after with: "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."
Downing Street later said Lineker's criticism of the new asylum policy was "not acceptable".
Labour said comparisons with Germany in the 1930s "aren't always the best way to make" an argument.
Lineker, who has presented Match of the Day since 1999, is the BBC's highest paid star, having earned about £1.35m in 2020-21.
He has in the past been vocal about migrants' rights and has taken refugees into his home. He has also been critical of successive Conservative governments over issues including Brexit.
In October, the BBC's complaints unit found Lineker had broken impartiality rules in a tweet asking whether the Conservative Party planned to "hand back their donations from Russian donors".
The comment came after the then Foreign Secretary Liz Truss urged Premier League teams to boycott the Champions League final in Russia over the invasion of Ukraine.
The furore surrounding Lineker's latest remarks put pressure on the BBC, with director general Tim Davie having made impartiality a cornerstone of his leadership.
The broadcaster's editorial guidelines state that the organisation is "committed to achieving due impartiality in all its output" and that "public comments, for example on social media, of staff [or] presenters... can affect perceptions of the BBC's impartiality".
A spokesperson for the corporation said: "The BBC has social media guidance, which is published. Individuals who work for us are aware of their responsibilities relating to social media. We have appropriate internal processes in place if required.
"We would expect Gary to be spoken to and reminded of his responsibilities."
The corporation has also responded to previous criticism of Lineker by highlighting that he is not involved in its news or political output and is a freelance broadcaster, not a member of staff.