Several former England players have tipped the Irishman to replace Cook as skipper for the 50-over World Cup in Australia and New Zealand next year.
The much-maligned Cook is determined to continue in his role and Morgan, who is vice captain to the opening batsman, has become the latest of his teammates to come out and publicly back his skipper.
"Cooky has been exceptional this summer," Morgan told reporters. "He's been under the pump quite a lot - and in challenging times in English cricket at the moment, he's come out really well.
"He's been outstanding dealing with it. I have every confidence in him. I have a huge amount of respect for him, both as a player and as a captain - and I look forward to him leading the side out come Sri Lanka and come the World Cup."
Morgan will captain England in their Twenty20 international with India at Edgbaston on Sunday.