Williams, 21, is under contract with the Merseyside club until the end of 2024.
The right-back is unlikely to get first-team opportunities at Liverpool due to the presence of Trent Alexander-Arnold and would be expected to be the understudy to the England international.
He had an impressive last season with Fulham on loan and managed two goals and two assists in 14 appearances across all competitions.
This resulted in increased interest from multiple Premier League clubs which is understood to be the reason why the Reds have upped their asking price recently. Nottingham Forest had earlier agreed personal terms with the player but Liverpool’s fee demands have proved to be a major stumbling block.
Forest had a £12 million bid rejected for the player, while Fulham are also understood to be keen to sign the player. However, Southampton are Forest have surged ahead in the race and the former is readying a £20 million bid that might be accepted by Liverpool.
Liverpool will be happy with overall transfer dealings
The Reds have had a comparatively active transfer window thus far. Their biggest loss has obviously been Sadio Mane, who left for Bayern Munich. However, Liverpool fans will not be worried as the club have signed Darwin Nunez from Benfica and also have the likes of Diogo Jota and Luis Diaz capable of operating on the left wing.
Nunez, in particular, could prove to be a big signing for Liverpool as the team only required a primary goalscorer. Nunez has for obvious reasons been compared to Manchester City’s Erling Haaland and can be expected to hit the ground running at Anfield.
Liverpool have also signed highly-rated attacking midfielder Fabio Carvalho from Fulham, who can be expected to be a big player in the coming years. Finally, Neco Williams could be sold as the club has signed 18-year old right-back Calvin Ramsay for Aberdeen as well.
Liverpool have also worked hard to reduce their wage bill and have sold many players already, including Takumi Minamino who went to AS Monaco, and Divock Origi, who left for AC Milan.