The Premier League newcomers are hopeful they can agree a package of an initial £16m fee plus add-ons for the Welsh international.
Williams is relishing the chance to become a starter at the Premier League level, so personal terms should not be an issue.
Liverpool reportedly demand a package of £20m, but it looks like Forest are not too far away from meeting it considering the add-ons.
The Reds aim to sanction the move before they travel for the pre-season tour in the Far East.
Williams has been associated with Liverpool since joining the club's academy at the age of six. He has made 33 appearances for the first team, including 13 Premier League matches.