He said: "If you're telling me they're awful racist people, I would certainly apologise if you would like me to do that".
But pressed for an apology by Morgan, he said: "I don't want to cause any difficulty for your country".
"I know nothing about them".
"Perhaps it was a big story in Britain, perhaps it was a big story in the UK, but in the United States it wasn't a big story. I know nothing about them, and I know nothing about them today other than I read a little bit", Trump told Morgan.
Asked if he regretted retweeting the anti-Muslim videos, Trump told the British journalist: "It was done because I am a big believer in fighting radical Islamic terror". While apologising, he said he was the least racist person and he was not endorsing anyone.
During the 2016 presidential race, Trump campaigned for "a total and complete shutdown" of Muslims entering the United States on the pretext of preventing terrorist attacks. "Okay? I don't know anything about what you're even talking about with white supremacy or white supremacists ... you wouldn't want me to condemn a group that I know nothing about".
The US president then went on to say he has a "very good relationship" with British Prime Minister Theresa May and praised her for doing "a very good job".
May said in a statement that it was "wrong for the president to have done this", but she didn't rescind her offer for Trump to visit.
Morgan said that Trump "publicly apologised" for retweeting the messages and added that the President said he "did not know who they were".
Trump responded, "They are, but this was - I didn't do it".
"Heartfelt", tweeted one social media user.
Also, after white supremacist protesters rallied in Charlottesville, Virginia, past year and an anti-racism counter-protester was killed when a vehicle ploughed into a crowd, Trump said there were some "very fine people" among the protesters, even though they included KKK members and armed militia.
His chat with Morgan was the first worldwide TV interview with Donald Trump since he took office. "We will come to your defense should anything happen, which hopefully will never happen".
The piece with, a known supporter of, is the president's first global television interview since taking office and will air on Sunday.
"I support her, I support a lot of what she does and a lot of what she says".