Trump said repeatedly Friday that he knew "nothing" about the group's politics. He said the tweets showed his concern about the threat of radical Islamic terrorism.
"If you are telling me they're horrible people, horrible, racist people, I would certainly apologize if you'd like me to do that," the president told the ITV broadcaster.
Reaction in Britain was mixed to Trump's rare offer to admit a mistake. Many Britons noted that it wasn't really an apology; others said it was close enough.
At the World Economic Forum in Davos, Switzerland, the president appeared to be working to reset frayed relations with May and Britain, traditionally America's closest ally but now a country where Trump is so unpopular that governments on both sides of the Atlantic fear the mass demonstrations that would likely greet him on a visit to London.
Sitting beside May on Thursday, Trump dismissed the idea of division between the two as a "false rumor."
"We're on the same wavelength, I think, in every respect," Trump said. "And the prime minister and myself have had a really great relationship, although some people don't necessarily believe that, but I can tell you it's true."
Trump said he thought "the feeling is mutual from the standpoint of liking each other a lot." He and May are "very much joined at the hip" on security challenges, he added.
May was more circumspect. "We, too, have that really special relationship between the U.K. and the United States," May said. "It's at each other's shoulders. We face the same challenges across the world, and . . . we're willing to go and to defeat those challenges and meet them."
The optic was particularly problematic because many in Britain believe that in 2003, then-Prime Minister Tony Blair was led under false pretenses into a ruinous war in Iraq because of his overly chummy relations with President George W. Bush.
At that White House meeting last year, May invited Trump for a state visit to Britain, which traditionally would include pomp, parades and an audience with Queen Elizabeth II. The prime minister's office has said the visit is likely to come in the latter half of 2018.
But British politicians, including London Mayor Sadiq Khan and Labour Party leader Jeremy Corbyn, have said that Trump is not welcome.
Activists have predicted that Trump's appearance would spark the largest demonstrations in London since protests against the Iraq War in 2003.
In the television interview Friday from Davos, Morgan asked Trump what he thought about opposition to his visit by the likes of Khan and Corbyn.
"I don't care. I don't care. It's just one of those things. I don't say anything. You know why? I don't care," Trump said, adding: "The real me is somebody that loves Britain, loves the U.K. I love Scotland."
Trump listened as the interviewer described Britain First, which presents itself as a political party but is widely seen as an extremist group targeting Muslims, as "racist."
The president denied having any knowledge of the group when he shared three videos from Jayda Fransen, its deputy leader.
"Of course I didn't know that. I know nothing about them, and I know nothing about them today other than I read a little bit," Trump said. "I don't know who they are. I know nothing about them, so I wouldn't be doing that."
He added, "I am often the least racist person that anybody is going to meet."
After Trump retweeted the videos, Fransen expressed joy, tweeting "God Bless You Trump!" and noting that the president has 44 million Twitter followers.
Fransen was convicted of religiously aggravated harassment in November 2016 after abusing a woman wearing a hijab and was arrested a year later for a speech in Belfast that allegedly used "threatening, abusive, insulting words."
The videos showed alleged violent acts carried out by Muslims, including one of a teenage boy on crutches being beaten by a young man purported to be a "Muslim migrant." Dutch authorities said the assailant was actually born and raised in the Netherlands and was punished under Dutch law.
A statement from the British prime minister's office later said it was "wrong" to share such materials that promote "hateful narratives."
Trump, in turn, responded on Twitter. In a tweet that initially was directed at the wrong person, he wrote: "Theresa @theresamay, don't focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom. We are doing just fine!"
Many in Britain felt that Trump was almost gloating over terrorist attacks in London and Manchester.
They also pointed out that during the same period, there were repeated mass shootings and attacks in the United States by domestic assailants.
Schemm reported from Addis Ababa, Ethiopia.