Livesey, 24, missed the final four months of last season due to a knee problem but Abbott hopes to keep the defender at Brunton Park.
"That one wouldn't necessarily be a surprise because Paul Simpson [Shrewsbury boss] knows Danny well from his time here," Abbott told the News & Star.
"But I have to say that I've not heard anything from Shrewsbury or anyone else about Danny - but as a manager I would probably be the last to find out!
"He has been a big part of the football club's success over the last couple of years and not one I would readily want to lose. It would cost them a lot of money if it was the case because Danny is a player who has done really well for us."
On whether Michael Bridges will turn last season's loan deal into a permanent move, Abbott added: "Michael is in no rush to do anything - he is looking for the best offer for Michael Bridges.
He has other bits and bobs on the table but we are in with a shout.
"Within a week or 10 days that situation might resolve itself. We have to see what Michael wants to do. He needs to know where he's going, and we need to know as well."