Hammill, 21, made 14 appearances on loan last season and scored once, possibly one of the most vital goals last season when he equalised at Plymouth on the last day.
Davey told Barnsley World:
"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".