The former Real Madrid striker held a "raging" meeting with the club and its Argentinian coach Marcelo Gallardo last Friday, the source said.
Benzema told his coach and club management "he doesn't feel he is able to give his best because of the current pressure at Ittihad".
"Benzema asked to leave Ittihad temporarily to change the atmosphere," the source added.
Al Ittihad offered to loan the French striker to another Saudi Pro League club but he has refused, the source added.
Contacted by AFP, Benzema's entourage declined to comment.
The 2022 Ballon d'Or winner, who was one of the highest-profile signings of the Saudi league last year and is reportedly being paid 100 million euros (£85.6 million) a season, returned to the club 17 days late after its mid-season break.
He has since been training alone in Jeddah, according to the source, not allowed for "discipline reasons" to rejoin his team-mates who are at a training camp in Dubai.
Reports in England say Premier League club Chelsea would be keen to take Benzema on loan in the current transfer window.
His former club in France, Lyon, could not afford his wages, according to their spokesman.
He said a possible loan deal for Benzema "had never been studied or considered", adding: "It is not at all within Lyon's financial means."
A source close to the Saudi Public Investment Fund (PIF), which owns Al Ittihad and three other Saudi clubs, said anyway "there is no option that Benzema leaves this month".
Benzema has scored nine goals in the Saudi Pro League this season, but is well behind Cristiano Ronaldo, who has netted 20 times for Al Nassr.
Al Ittihad are seventh in the division after three successive defeats. They have scored the fewest goals and conceded the most of the four teams owned by the PIF.
The league is due to resume on February 7th.
Another high-profile recruitment to the Saudi league, former Liverpool captain Jordan Henderson, quit his club Al Ettifaq to join Dutch side Ajax last week.
Henderson insisted his decision to return to Europe was made for football reasons because he wants to be in contention for England's Euro 2024 squad.
His family reportedly found it hard to settle in the Gulf and his reputation as a supporter of LGBTQ rights was tarnished by his move to Saudi Arabia.
