McLeish admits he has already spoken to Downing about his future and plans to speak to the former Middlesbrough favourite again when the squad returns for pre-season training.
"I understand why there's teams interested in him and that is what happens when you play well consistently," said McLeish.
"His performances last season were top drawer. That's why he's not for sale. The club don't want to sell him.
"I spoke to him briefly last week and I intend speaking to him again when all the lads are back in for training."