Van Persie was stretchered off late in the day against Olympiakos on Wednesday, with the 30-year-old having scored a superb hat-trick to send United through to the quarter-finals of the Champions League.
Speaking immediately after the game David Moyes confirmed the striker had taken a knock on the knee but Kluivert is confident that it is not serious and that he could still be fit to feature in the crunch meeting with title-chasing Manchester City.
He told Algemeen Dagblad: "Robin went right to the hospital on Thursday morning. His upper leg was quite stiff. We have to wait for the results, but we have high hopes that it's not a heavy injury. Maybe he can play against Manchester City next week."
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He added: "It was a wonderful match vs Olympiakos. United really had to win to save their season. If you see them start like that, with a lot of pressure going forward, it's really impressive. Really nice to see as well.
"I have a lot of contact with Robin and I like watching him play. He deserved this. He was really keen on a good match on Wednesday. A few days earlier they got trashed by Liverpool, so they needed to answer, and that's just what United did, with Van Persie starring."