The skipper's deal runs out at the end of the season and Brendan Rodgers has told the Foxes board he wants him to stay.
Morgan, who led the club to their shock 2016 title win, has played 21 times this term including starting Rodgers' first game, a 2-1 defeat at Watford on Sunday.
Rodgers said: "I've spoken to the club and I would like him to get a year. He is ready if needed and of course he understands the position he is in - that he won't start every game. It happens to the best."
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