The Canaries first tried to sign the Scotland international last summer, but their offers were turned down by the Championship club.
Despite a change of manager, with Chris Hughton now in charge at Carrow Road, their interest in Snodgrass has remained.
They have made a fresh bid for the 24-year-old, and Sky Sports understands Norwich have tabled two offers in the last week for the Scot.
With Paul Lambert, who is now at Aston Villa, among a host of confirmed admirers in the Premier League, Norwich will be hopeful they can strike a quick deal to land the in-demand Scot.
Snodgrass has just 12 months left on his current deal at Elland Road, having failed to agree a new contract with Neil Warnock's side.
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