Morgan, who is now 37, came on late in the Foxes' 1-0 win over Chelsea at Wembley on Saturday.
The centre-back was part of the Premier League winning Leicester side in 2016 as well.
Morgan said about his future after the game: "Nothing's been announced yet, I need to discuss it with the club, but I don't think I've got too many more miles on the clock, we'll say that.
"I wasn't sure if I was going to make the squad. The gaffer asked me if I could make it on the bench and I said if I'm needed for the last 10 minutes, I'll be available.
"We were 1-0 up and the last thing I expected a week ago was to be playing again, but I came on and now I've won an FA Cup."