BBC Sport reports City is now the favourite in the hunt for the 23-year-old ahead of this summer's transfer window.
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Manchester United is also interested in the player, who seems to be at the top of the wish list.
United have also been linked with Carlos Baleba (Brighton), Sandro Tonali (Newcastle United) and Bruno Guimarães (Newcastle United), among others.
Finally, Bayern Munich is also interested in Anderson, but it may be difficult to out-compete the English clubs.
Forest is expected to demand £80m to sell Anderson this summer.
Anderson's contract with Forest runs until the summer of 2029.