The Only Way Is Essex star is said to be concerned that associating with Tweed, who was married to the late Jade Goody, could harm his career.
"Mark was put in a real Catch-22 situation," a source told The Mirror. "Jack's been his best mate since they were little kids, but after the pair got in a bit of trouble in January, Mark was really shaken-up.
"He realised he was in danger of losing it all if he didn't clean up his act. In addition, some of the brains behind the programme think Jack's reputation is a bit dodgy for their clean-cut image.
"For the first series they were extra careful to keep Jack off camera when they were filming at their flat. But now the show has taken off big time and the pressure is on Mark to look out for his image. He's becoming a big star and TV bosses don't want anything to damage that."
The report adds that Tweed is "p**sed off" with Wright because they have been friends since childhood.
Wright recently denied claims that his relationship with Lauren Goodger is fake.