Lineker said the language in which the plan was set out was "not dissimilar to that used by Germany in the 30s".
The BBC has impartiality guidelines and the corporation said Lineker was being "spoken to" about his responsibilities.
Home Secretary Suella Braverman said she was "disappointed" by the comments.
On Tuesday, the government outlined its plans to effectively ban anyone arriving via an illegal route from claiming asylum in the UK.
Anyone found to have entered the country illegally will also be blocked from returning or claiming British citizenship in future.
The measure is part of attempts to address an increase in the number of people arriving in the UK via Channel crossings each year, which rose from around 300 in 2018 to more than 45,000 in 2022.
Responding to a video message setting out the policy by Ms Braverman, Lineker tweeted: "Good heavens, this is beyond awful."
Told by another user he was "out of order", he added: "We take far fewer refugees than other major European countries.
"This is just an immeasurably cruel policy directed at the most vulnerable people in language that is not dissimilar to that used by Germany in the 30s, and I'm out of order?"
It is not clear which language in particular Lineker was referring to, but Ms Braverman's video and accompanying tweet included the words "enough is enough" and "we must stop the boats".
Following the comments, the home secretary 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."
The decade saw the rise to power of the Nazi party in Germany and persecution of Jewish people, leading to the Second World War.
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.
In 2018, after BBC cricket correspondent Jonathan Agnew called on him to "keep your political views... to yourself", Lineker responded: "I'm the face of my own Twitter account. I'll continue to tweet what I like and if folk disagree with me then so be it."
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".
Mr Davie said he had not spoken to Lineker.
Asked about how many "strikes" the presenter has had over social media posts, he said: "I wouldn't talk specifically about individuals, I don't think it's right.
"I think the BBC absolutely puts the highest value on impartiality and that's clearly important to us."
* People removed from the UK will be blocked from returning or seeking British citizenship in future
* Migrants will not get bail or be able to seek judicial review for the first 28 days of detention
* There will be a cap on the number of refugees the UK will settle through "safe and legal routes" - set annually by Parliament
* A duty on the home secretary to detain those arriving in the UK illegally and remove them to Rwanda or a "safe" third country - this will take legal precedence over someone's right to claim asylum
* Under-18s, those medically unfit to fly, or those at risk of serious harm in the country they are being removed to will be able to delay removal
* Any other asylum claims will be heard remotely after removal
Conservative Party deputy chairman Lee Anderson said Lineker was "out of touch" and should stick to football.
Writing on Twitter following Lineker's comments on Tuesday, he said the presenter had "piped up again with his virtue-signalling nonsense".
"This is just another example of how out of touch these overpaid stars are with the voting public," he said. "Instead of lecturing, Mr Lineker should stick to reading out the football scores and flogging crisps."
Speaking to the Daily Telegraph, Conservative MP Craig Mackinlay said Lineker's comments were "foul, ill-conceived and disgraceful" and called on the BBC to sack him.
Shadow home secretary Yvette Cooper told BBC Radio 4's Today programme she did not agree with the comment, adding it is "entirely" a matter for the BBC when asked whether it should take significant action against him.
She continued: "He's somebody that's spoken out very strongly on lots of different issues, and people who feel strongly should be able to speak out and say the things that they feel."
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.