Chester are also hoping to extend winger Robbie Foy's loan deal until the end of the season. Foy has been at Chester a month and has impressed Rush and his backroom staff.
Chester chairman Stephen Vaughan admits that he has spoken to Rick Parry about both Foy and Harrison.
"We have contacted Liverpool's chief executive Rick Parry about Robbie, and we hope they will agree to the loan extension," said Vaughan.
"Robbie has given the squad extra options during his period with us, and everybody at the club has been impressed by the contribution he has made.
"Paul is a player of great promise, and he would be a valuable acquisition for the future if we could sign him."