Speaking at Boston Celtics media day, O'Neal, 38, was asked if he had heard Bryant, 32, bragging about having one more ring than him in the moments immediately after winning another championship in June.
"Yeah I heard it," O'Neal said. "My whole career I've been the measuring stick. Glad to see I'm still relevant.
"I would've been more upset, more hurt, if Tim Duncan would've made the comment. I don't compete with guards. They have the ball more than I do, they shoot way more than I do. You can't compete with guards.
"I'm only competing with Tim Duncan so hopefully I can get number five before Tim Duncan, respectfully before Tim Duncan."
San Antonio's Duncan and O'Neal each have four NBA championship rings, while Bryant and Derek Fisher lead active players with five.
O'Neal, who was first drafted by the Orlando Magic way back in 1992, signed with the Celtics in August.