Jonny Evans could sign a new deal with Leicester before the end of the year.
The defender's current contract expires in the summer but talks over a fresh two-year offer are at an advanced stage.
Boss Brendan Rodgers said: "I'm hearing from the club it's pretty close. We're very hopeful very soon we'll be able to finalise that agreement.
"It could be sooner than that (January), we'll just wait and see. I know the representatives of Jonny and the club are very close.
"If it's this year, great, if it's the new year hopefully it'll be very early."
Rodgers had always been confident that the three-time Premier League winner would sign a new deal.
Evans, 32, joined from West Brom for £3.5million in 2018 and has played 87 times for the Foxes, scoring twice.
He returned from suspension to help Leicester beat Tottenham 2-0 on Sunday and lift them to second in the Premier League.
Evans' imminent new deal is further good news with the club due to move to their new £100million training ground this week.
The first team will train at the facility, which has a private nine-hole golf course, for the first time on Christmas Eve.
Rodgers added: "The players have an amazing recreational area, they can sit outside and have their tea and coffee.
"There are rooms for them so when they train in the morning they can have their lunch and then have a sleep and train in the afternoon.
"There are analysis hubs where they can analyse their game and an indoor area – and if the weather is nice and they're allowed they can have a bit of golf."
