The 21-year-old was initially expected to complete a permanent move to Fulham after playing a major role in the Cottagers' return to the Premier League last season.
Williams recorded two goals and two assists in 14 Championship games after joining on loan in January, but Marco Silva's side did not agree a deal to sign the Wales international.
Williams was therefore due to return to Liverpool ahead of the 2022-23 season, but the Reds have brought in Calvin Ramsay to challenge Trent Alexander-Arnold for the right-back spot.
With Joe Gomez also not expected to leave the club in the coming weeks, Liverpool have been open to letting Williams leave for the right price this summer.
Nottingham Forest had been frustrated in their efforts to sign Williams, but The Express now reports that the Tricky Trees have struck an agreement to sign the right-back for a £17m fee with add-ons.
The report adds that Liverpool initially rejected a deal worth £15m from Steve Cooper's side, which would have seen the Reds pocket an initial £11m with a potential £4m in bonuses.
However, the two clubs have now reached an agreement over a fee, and there are not expected to be any issues with personal terms before Williams undergoes his medical.
The Anfield academy graduate - who has been on the books on Merseyside since 2009 - prepares to leave Jurgen Klopp's side after recording six assists in 33 games for the club in all tournaments.
The defender has won the Premier League, Club World Cup and EFL Cup since breaking into the Reds' first team, but his opportunities in 2021-22 were few and far between.
Williams made eight appearances before being shipped out to Fulham and played just seven minutes of Premier League football, coming off the bench in October's 5-0 thrashing of Watford.
Forest have already made a number of acquisitions since returning to the Premier League, welcoming Taiwo Awoniyi, Moussa Niakhate, Dean Henderson and Giulian Biancone to the City Ground.
As well as adding Williams to their right-back ranks, the newly-promoted side are also said to be in discussions with Bayern Munich over a deal for left-back Omar Richards.
The Tricky Trees' first game back in the top flight sees them travel to Newcastle United on August 6, while Liverpool do battle with Fulham at lunchtime on the same day.