The six-year-old came close to gaining a first win since he landed a Group Three at Newcastle in June 2015 when second in the Stewards' Cup at Goodwood last Saturday.
Aeolus finished ninth at Ayr last year and Walker believes a second crack at the six-furlong contest is a logical goal.
Walker said: "It was a great run in the Stewards' Cup. It was just frustrating to finish second as it was a long-standing target.
"It was a case of so close, yet so far, but I'm sure he will have his day.
"I think he is an Ayr Gold Cup horse. He ran OK in it last year when he was drawn badly and I think with a good draw he would be one of the favourites.
"He just travels so well and the cheekpieces helped as well at Goodwood. I think the Ayr Gold Cup would be the perfect race for him.
"It is just a case of whether we try to protect his mark as he has not been put up for Saturday, which is surprising but pleasing, or have a run in another handicap.
"If he is only going to win one handicap this year, though, I would rather it be a big one."
The Lambourn handler is considering stepping last year's Craven winner Stormy Antarctic back up to a mile following his no-show in the Lennox Stakes at Goodwood last week.
He said: "It has been continually frustrating with him. It really is just back to the drawing board.
"There are no plans for him at the moment. We have tried to lower his sights in two Listed races this year, but he has come up short in both and there is only so much lowering you can do with a horse rated 111.
"I don't think he is a seven-furlong horse, I think he probably needs to go back up to a mile as he settled well enough at the Curragh and Goodwood.
"I just want to get him back winning again."