Van Persie has been the subject of much speculation in recent weeks, with United, Manchester City and Italian giants Juventus all linked with the Dutchman after he announced that he would not be signing a new contract with the Gunners in early July.
However, no club has been able to agree a fee with the north London club, and the prospect of the 29-year-old remaining at the Emirates Stadium for another year has begun to look an increasingly likely one after the Gunners' signing of Santi Cazorla.
Speaking after United's pre-season penalty shoot-out defeat to Barcelona, he said: "We have made a bid and they've been trying to negotiate with other clubs. We just have to persevere.
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"Hopefully it will come our way. We're trying our best but there's no progress at this moment in time."
Sir Alex also claimed that the Gunners' unwillingness to sell their captain was proving a stumbling block.
He added: "I don't have a gut feeling on it at the moment, I must admit. We're not getting any breakthrough with Arsenal.
"It's difficult to say why they're operating this way. I don't know what their thoughts are because they're not giving anything away."