The 21-year-old Scouser impressed highly on loan last season and scored an important goal on the last day of the season against Plymouth.
Davey told Barnsley Player: "We have been in conversation with Liverpool again this morning. I am yet to find out the outcome of that but we have made a bid for Adam today.
"We have put a bid in for a permanent deal. Hopefully that will be a successful one and we can get Adam into the football club as soon as possible. "There have been positive noises coming from Anfield and I do not think that we are that far apart in our valuations so hopefully we can come to an agreement."