The classy pair met in the Coral-Eclipse at Sandown in July, where Aidan O'Brien's French 2000 Guineas and Derby winner came out on top in impressive fashion, with Mishriff edged for second by Addeybb.
St Mark's Basilica was in receipt of 10lb that day - and the John and Thady Gosden-trained Mishriff again had to concede weight when clashing with Derby hero Adayar in the King George VI And Queen Elizabeth Stakes at Ascot on his latest run.
Beaten just under two lengths over a mile and a half, Egan was pleased with the run but expects Mishriff to be better suited by returning to 10 furlongs on the Knavesmire, with the three-year-old allowance reduced to 7lb, too.
He said: "I think the plan is the Juddmonte, dropping back to a mile and a quarter should play to his strengths.
"I thought he ran a fantastic race over a stiff mile and a half at Ascot, but probably just got outstayed by a horse giving him 11lb.
"Obviously we probably won't be taking on the Derby winner in the Juddmonte, but maybe we might take him on again later in the year.
"It's an exciting feeling, we'll probably have to take on St Mark's Basilica. Mishriff was a little bit ring-rusty the first day at Sandown, so it would be nice to have a go at him again."
Mishriff struck Group One gold over 12 furlongs in the Dubai Sheema Classic earlier in the year, adding his huge haul in the Saudi Cup previously, and Egan hailed the Make Believe colt's adaptability.
He added: "He's such a versatile horse. Over the mile and a half you have to ride a bit more of a race on him to make sure he relaxes and settles early because he has got a good turn of foot. He showed that turning into the straight at Ascot, he loomed up on the outside looking like he was going to win the race.
"He's a very versatile horse with a lot of attributes that a Group One horse like him needs."