In response, the actor called on him to work for all Americans "as aggressively as you worked for your ratings".
Monday`s season launch was seen by an estimated 4.9 million people - down 43% on the last season premiere in 2015.
Schwarzenegger has received mixed reviews for his debut as the new star of the show.
The veteran action star and former California governor has replaced Mr Trump`s "You`re fired" catchphrase with "You`re terminated" - a reference to his role in The Terminator film and its sequels.
`I wish you luck`
In the tweets, sent on Friday, Mr Trump wrote: "Wow, the ratings are in and Arnold Schwarzenegger got "swamped" (or destroyed) by comparison to the ratings machine, DJT.
"So much for being a movie star - and that was season 1 compared to season 14. Now compare him to my season 1."
More than 11 million people watched the opening episode of Celebrity Apprentice in 2008, according to Variety.
Referring to Republican Ohio governor John Kasich and Democratic presidential candidate Hillary Clinton, Mr Trump added: "But who cares, he supported Kasich & Hillary."
In return, Schwarzenegger tweeted: "There`s nothing more important than the people`s work, @realDonaldTrump."
He added: "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."
Mr Trump helped create the show and starred on it until 2015, when his political career took over.
The contestants on the current series, who compete to raise money for charity, include boxer Laila Ali, Boy George and Motley Crue singer Vince Neil.
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