The 21-year-old enjoyed a productive three-month loan spell at Oakwell last season and the Tykes are now keen to secure his services on a permanent deal.
Barnsley have now tabled a bid for Hammill and Davey is hoping that their bid meets the Anfield outfit's valuation.
He 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."