She told a podcast she had also heard producers on an unnamed production say "too many brown people" would not sell.
Nagra made her name in the 2002 UK film Bend It Like Beckham before appearing the American medical drama series ER.
While noting how on-screen diversity had improved, she said she hoped it was not just a quota-filling exercise.
"We're having the conversations happening more and more and things have moved on," she told the Celebrity Catch Up podcast.
"My fear is that - which I think was happening a few years ago - it became about, 'Oh well let's get this box ticked'," continued the actress, who lives in Los Angeles.
"I remember asking to go for a job because an actress basically left - it was a well-known TV show here. I remember phoning my agent and I went, 'Do you think maybe you could just suggest me? The character is very non-specific in terms of family and between 35 and 40 [years of age].
"And the word that came back was that they've already got an Indian person on the cast.
"I went, 'Yeah but I'm completely different to that person'. Is that ever gonna happen when you say that, 'No we've already got a white person on the show'?
"I don't think that conversation is happening - so it's just very hard."