Spence had a successful stint on loan at the City Ground last term as he helped Nottingham Forest to clinch promotion to the Premier League.
Forest have been keen to sign Spence on a permanent basis from Middlesbrough, but faced fierce competition for his signature.
Now they are switching their focus away from Spence, who is wanted by Tottenham, and towards Liverpool’s Williams.
Spence is talking to other clubs and his price is now rising, pushing him increasingly out of reach for Forest.
Nottingham Forest could pay Liverpool a fee of £15m to take Williams to the City Ground.
Boss Steve Cooper rates Williams, who is aware that Liverpool are planning to sell him this summer.
The 21-year-old Wales international came through the youth set-up at Liverpool and spent the second half of last season on loan at Fulham, helping the Cottagers to promotion from the Championship.
Tottenham have held talks with Middlesbrough for Spence, but have yet to find an agreement.
