The Republic of Ireland star was impressive in a 1-0 Nations League win over Finland on Thursday.
While the Republic were then beaten by England days later, Kelleher earned praise for his game against the Fins.
"Caoimhín is getting regular games. I was working with Martin (O'Neill) when we brought him in (to the Ireland set-up). We liked him straight away from the first session," Keane told ITV Sport.
"We had Seamus McDonagh saying 'we like everything about him. He makes the other keepers look lazy'. Seamus said he had a great temperament.
"When you play for the big clubs and play in big games - he's made one or two mistakes but he takes it in his stride.
“He's getting the run of games now at Liverpool and I think he's got a great chance. I like everything about him and the big bonus is he's from Cork!"