Livesey made 38 League One appearances for Carlisle last season
Carlisle United boss Greg Abbott has offered contracts to Danny Livesey, Tom Aldred and Andy Cook.
Both defender Aldred and striker Cook, who has has been on loan at Barrow this season, were offered one-year deals.
Livesey, a defender, has been offered a two-year contract but all three are yet to commit their futures at the club.
"We have had some very positive talks with Danny and think we might hear something from him by the end of the week," Abbott told the club website.
He added: "Tom and Andy are away on holiday so we would expect to hear back from the at some point in the next fortnight."
I'm accepting in this instance that Richard Keogh wants to move on, but I wish him well in the future, as he is an excellent lad to have around
Greg Abbott
The Carlisle boss added that he was hoping to sign Adam Clayton, who spent last season on loan at the club from Manchester City.
"We're going to speak to Adam and his agent face to face and see where we are with that," Abbot said.
"I'd prefer to do that as a permanent deal rather than a loan but that all depends on Manchester City."
Abbott will also have to plan for next season without popular defender Richard Keogh, after he turned down the offer of a new deal at Brunton Park.
Keogh, 23, had been at the club since joining from Bristol City in November 2007.
Abbot said: "This is a hard one to accept, if I'm honest, because he brings a lot into the dressing room, not just as a player but in the way he conducts himself.
"His influence on the other players, on the backroom staff and on the fans has been tremendous and I honestly don't think that we could have done anything more in terms of the package that we offered to persuade him to stay.
"Sometimes as a manager you have to accept defeat. I'm accepting in this instance that Richard wants to move on, but I wish him well in the future, as he is an excellent lad to have around."
Among the other players to leave the club this summer are full-back Evan Horwood, defender David Raven, striker Scott Dobie and goalkeeper Lenny Pidgeley, who have all been told they will not receive new deals when their current contracts expire this summer.
Bookmark with:
What are these?