Foot Mercato says Forest have made a €30m offer for the England international.
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However, City immediately rejected the proposal, insisting the Forest bid falls well short of their valuation.
Lewis, who can play in midfield and at fullback, is viewed at City as potentially former captain Kyle Walker's natural successor at right-back.
Forest, meanwhile, are also finalising terms with City for Lewis' fellow City academy graduate, James McAtee.