"Arsenal always wants to play Champions League and I think Arsenal needs to play Champions League," said van Persie, signed from Feyenoord in 2004.
"If you have players like this, good players, big stadium, then I think you need to play in the highest league in Europe.
"But this is football - if you don't win two games then you don't qualify and we play UEFA Cup."
"I know them quite well as I have Dutch satellite," he said.
"It was quite, though, a big shock because in Holland everyone expected Ajax to win and in the end Twente played better and these things happen in football.
"They had a great year last year, but they lost two of their best playmakers - Karim El Ahmadi and Orlando Engelaar. However, they got Theo Janssen from Vitesse, so they are still in good shape with a good team."
Van Persie feels Twente will have been disappointed to have drawn such a big name as Arsenal in the qualifiers, even though McClaren insists all the pressure is on the English team.
And the Gunners forward claims the much-criticised former national team head coach has been well received in Holland.
"They are really happy with him and almost proud to have him, because it was quite a strange move to come from England manager to Twente," he said.
"But they are all saying they really like his style of coaching."