The Anfield academy graduate spent the final two and a half months of last season on loan at Oakwell after a spell at Blackpool earlier in the campaign.
Davey is keen to sign him on a permanent basis but revealed that Liverpool are currently assessing Hammill before making any decision on his future.
Davey told the club's official website: "We have made an offer for Adam and were told that Liverpool wanted to have a look at him through pre-season.
"Our offer is still on the table but we cannot just wait around and we have to make sure we have other targets in mind if that answer is a negative one.
"On the other hand it is not going to blow our budget if we get the right answer on Adam two days before the start of the season."