The Turkish sports newspaper reports that the Yellow Canaries are preparing an offer for the Belgium international over the winter transfer window.
The Yellow Canaries currently have two strikers following the sale of Moussa Sow and Dirk Kuyt over the summer.
Fenerbahçe did bring in Liverpool youngster Lazar Markovic on a one-year loan but want an additional striker to add depth.
Liverpool reportedly turned down a loan offer from Fenerbahçe for Origi over the summer but could be tempted into parting ways with the young striker in January as he has fallen down the pecking order behind Ings, Benteke and Firminho.
The 20-year-old has played just 88 minutes of football this season including only 16 minutes in the Premier League.
The Reds signed Origi on a £8.84 million transfer from Lille last summer but he was loaned back to the club he first made his name at.