The Premier League champions want the Northern Ireland international as cover for Andy Robertson and submitted the offer, which includes a sell-on clause, on Thursday.
Norwich value the 22-year-old at closer to £20m and immediately rejected a deal that would have made Lewis Liverpool’s first summer signing.
Lewis is understood to be keen on a move to Anfield, where he accepts he would be back-up for the Scotland captain. Robertson has rarely missed a Premier League game over the past two seasons but Jürgen Klopp is wary of relying exclusively on James Milner to provide cover in an intense and truncated 2020-21 season.
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