Donald Trump says his replacement on The New Celebrity Apprentice television show is to blame for its poor ratings, and not, as Arnold Schwarzenegger says, Trump's continued involvement with it.

Schwarzenegger, who replaced Trump as host of the show while the president remained its executive producer, announced on Friday that he is through.

He said even if he were asked again, he would not want to host the show.

"With Trump being involved in the show, people have a bad taste and don't want to participate as a spectator or as a sponsor or in any other way support the show," he said in an interview with the Empire website.

"It's a very divisive period now, and I think this show got caught up in all that division."

But Trump, who has taunted Schwarzenegger for the show's poor performance in the ratings and even mocked him during an address at the National Prayer Breakfast, tweeted in response on Saturday: "Arnold Schwarzenegger isn't voluntarily leaving the Apprentice, he was fired by his bad (pathetic) ratings, not by me. Sad end to great show."

The show wrapped up its 15th season last month.

It averaged fewer than five million viewers per episode. That was lower than any other incarnation except for a 2010 Trump-hosted edition without celebrities, and the most poorly-rated edition among the youthful viewers NBC cares most about, the Nielsen company said.

NBC had no immediate comment about whether the show would return with a new host.

The former California governor said he would not mind working with NBC and producer Mark Burnett again, "on a show that doesn't have this baggage".

Shannon Coulter, co-founder of the #GrabYourWallet Trump protest, agreed with Schwarzenegger.

The effort aims to spur consumer boycotts of companies doing business with the Trump family, including sponsors of Apprentice.

"My impression is that The New Celebrity Apprentice's poor performance has everything to do with Donald's continued presence and the accompanying 'baggage' Mr Schwarzenegger mentioned," she said.

"We applaud Mr Schwarzenegger's decision to leave the show and to distance himself from the toxic Trump brand."

After Schwarzenegger's debut in January, Trump used Twitter to report the show had been swamped "in comparison to the ratings machine, DJT", using his initials.

Schwarzenegger shot back on social media that he hoped Trump would work for the American people as aggressively as he did for his ratings.

The actor, like Trump a Republican, did not endorse Trump and said during last year's campaign that he would not vote for him.

When The New Celebrity Apprentice ended its season last month, Trump tweeted that Schwarzenegger "did a really bad job as governor of California and even worse on The Apprentice, but at least he tried hard".