Their relationship has been described as "unsalvageable," with Paul demanding a trade following Houston's elimination loss to the Golden State Warriors and Harden issuing a "him or me" ultimatum to Rockets management, league sources told Yahoo Sports' Vincent Goodwill.
"It can't be fixed," another source told Goodwill.
Paul commented on the rumors on an Instagram post from Bleacher Report, writing "Damn! That's news to me..."
Harden and Paul went almost two months without speaking to each other during the regular season and Harden has not responded to Paul's repeated attempts to contact him this offseason, Goodwill reports, citing sources.
But Rockets general manager Daryl Morey says Paul has never asked for a trade and denies any issues between the All-Star backcourt partners, according to Jonathan Feigen of the Houston Chronicle. Morey claims he and the pair have spoken together several times this offseason regarding the club's plans for free agency.
Paul and Harden got into a verbal on-court exchange about ball distribution during Houston's season-ending loss, an encounter that reportedly continued postgame off the court.
Harden has also been annoyed by Paul's personality and extended periods out of the lineup, sources told Goodwill.
"Chris doesn't respect James' standing in the league, and James doesn't respect the work Chris has put in to this point," one source said.
Acquired from the Los Angeles Clippers in the 2017 offseason, Paul nearly led the Rockets to the NBA Finals that season. But an assortment of injuries has forced the guard to miss 48 regular-season games and two pivotal postseason contests - Games 6 and 7 of the 2018 Western Conference finals - over his past two campaigns.
Dealing the 34-year-old could prove difficult; Paul is owed more than $79 million over the next two seasons and has a $44-million player option for the 2022-23 season.
Paul averaged 15.6 points, 8.2 assists, and 4.6 boards across 58 appearances this past season, marking his lowest scoring average since the 2010-11 campaign.