The Daily Star says Van Persie went into his old dressing room after the 1-1 draw at The Emirates to say he was convinced the Gunners would join him in the Champions League next season.
An Arsenal source said: "Robin said how well we played, especially in the first half, and that it was the toughest game United had had for a while.
"He reckoned it was the best he had seen us play and that we would qualify for the Champions League for sure.
"Despite the criticism from the fans, Robin's well-liked at the club.
"There's a lot he could have said about why he moved to United, but he kept quiet before and after the game and was also very respectful when he scored the goal, not celebrating.
"But he's happy now after finally winning something and he had a big smile on his face."