Las Vegas-based Bovada has listed Rose and McIlroy as the joint favorites for Royal Liverpool at 14-to-1. They are followed by Adam Scott and Henrik Stenson -- Nos. 1 and 2 in the world -- at 16-to-1.
Tiger Woods is next at 18-to-1.
Woods is a three-time Open champion who won at Royal Liverpool in 2006. But he missed three months because of back surgery, and he missed the cut by four shots when he returned three weeks ago at Congressional.
Looking for a long shot? Look no further than Nick Faldo at 1,000-to-1.