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In an interview to ITV's "Good Morning Britain" show taped in Davos, Trump said he knew "nothing" about the Britain First group before sharing three videos by its deputy leader, Jayda Fransen, in November.
"If you are telling me they're horrible people, horrible, racist people, I would certainly apologise if you'd like me to do that," he told presenter Piers Morgan in the interview that will be aired on Sunday.
Trump said the controversy was not a big story in the US and said he retweeted the videos "because I am a big believer in fighting radical Islamic terror".
One purported to show a group of Muslims pushing a boy off a roof.
Another claimed to show a Muslim destroying a statue of the Virgin Mary and another claimed to show immigrants hitting a Dutch boy on crutches.
"I know nothing about them, I don't want to be involved with people (like that)," Trump told Morgan.
The President's sharing of the controversial videos sparked a Twitter row with UK Prime Minister Theresa May last year, after her spokesman said it was "wrong" to share the inflammatory posts with his followers.
Trump told May to focus on "terrorism" in Britain instead. He subsequently pulled out of an expected visit to Britain to open the new US Embassy building in London, leading to speculation of a diplomatic rift.
However, after meeting May at the World Economic Forum in Davos, Switzerland, on Thursday, he insisted that relations were in good shape. "We actually have a very good relationship, although a lot of people think we don't."
Trump told May: "We love your country."
He also said the US and UK were "joined at the hip" on military matters, while May said they stood "shoulder to shoulder" in facing shared threats.