The movie is a biopic about astronaut Neil Armstrong, the first man to walk on the moon, with Ryan Gosling starring in the lead role.
It has caused controversy in America because the film does not feature the moment the country’s flag was planted on the moon, prompting the US president to say he would not be watching it.
“It’s almost like they’re embarrassed at the achievement coming from America,” Mr Trump told the Daily Caller.
Explaining his decision not to include the flag-planting in First Man, Chazelle said: “It was not a political statement, it was merely an aesthetic choice.
“I was surprised by all of that. It felt to me that the point of making a movie about such a famous event, with such famous imagery associated with it, would be not to just reproduce that imagery but to look at what happened in between the story.”
The Crown star Claire Foy, who plays Armstrong’s wife Janet in First Man, said people should make up their own minds about the movie.
She told Sky News: “I do think it’s interesting that the conversations that are happening are largely by people who haven’t seen the film.
“America and Americanisms are very prevalent throughout, so people should really go and see it.”
Chazelle, who won the Academy Award for best director in 2017 for La La Land, said he was keen tell Armstrong’s story because so little is known about his personal life.
“It was actually right before La La Land that I sort of started toying with this,” he said.
“I read Jim Hansen’s book about Neil, that the movie is based on. It was just a fascinating story.
“(I realised) that there was so much to such a famous event that I didn’t know… and how little the world knows about the moon landing and about Neil and Janet.”
Chazelle, who at the age of 32 became the youngster person to ever win best director at the Oscars, said he wanted to peel back the mythology of the time period of the Apollo 11 mission.
“They were such private people so you got this simple heroic image when you thought about them and that period in American history,” Chazelle said.
“And it was all gilded and I didn’t realise they had suffered and the tragedies and ordeals they went through and they tried to solider through together as a family.
“It felt like it was a remarkable story about fallible human beings being thrown into a completely extraordinary circumstances and doing the best they can and so out of that came this iconic event.”
Foy, who recently won an Emmy for playing The Queen on Netflix series The Crown, said she was desperate to work with Chazelle.
“I was really interested to know why he wanted to tell the story,” she said.
“To me I was like you’ve made Whiplash, La La Land and now you’re making a film about the moon landing but as soon as you start talking about it, with a man at the centre of it, the sacrifice of not just him but of his family… I was just begging to do it.
“Janet herself was an extraordinary person, an amazing woman but at the same time all those women (the astronauts’ wives) were and that’s what I thought was brilliant about the story.”
First Man is out in the UK on 12 October.