Former action movie star Schwarzenegger responded by suggesting that Trump turn his attention to his work as the nation's new president. He quoted Civil War-era President Abraham Lincoln in advising Trump.
Trump, who became a household name as the tough host of "The Apprentice" and sister show "The Celebrity Apprentice" for 14 seasons, slammed a 43 percent fall in the 18-to-49-year-old audience for the new TV show hosted by Schwarzenegger, which premiered on NBC earlier this week.
"Wow, the ratings are in and Arnold Schwarzenegger got 'swamped' (or destroyed) by comparison to the ratings machine, DJT," tweeted Trump, who controversially retains an executive producer credit on the reality show.
"So much for being a movie star - and that was season 1 compared to season 14. Now compare him to my season 1," he added.
Schwarzenegger, a Republican who served two terms as California governor but who has said he did not vote for Trump, responded swiftly.
"I wish you the best of luck and I hope you'll work for ALL of the American people as aggressively as you worked for your ratings," the "Terminator" actor tweeted.
He then asked Trump to study a quote from Lincoln's inaugural address in 1861, about bringing Americans together, saying he hoped it would serve as an inspiration.
"We are not enemies, but friends. We must not be enemies," the quote begins.
Some 20 million Americans were regular watchers of "The Apprentice" in the show's first year, falling to around 6 million in 2015 for the last episode of "The Celebrity Apprentice" hosted by Trump.
The TV audience for Monday's revamped show with Schwarzenegger in the boardroom was 11 million, with a larger, 43 percent, drop-off in the 18-49-year-old demographic coveted by advertisers compared with the show's last premiere in 2015, Nielsen data showed.
Trump will be inaugurated as U.S. president on Jan 20.
Banner and thumbnail credit: Reuters, Jonathan Ernst