The report claims that the Yellow Canaries have entered talks with Twente and that they want to wrap up the transfer for €7 million.
Twente reportedly want €9 million for the 23-year-old but Fenerbahce do not want to pay more than they sold Luis Nani for.
The Portuguese intentional will join Valencia on an €8.5 million transfer fee after the end of Euro 2016.
Ziyech has 39 goals and 37 assists in 100 Eredivisie performances and does have another two-years left on his current contract so Twente are in no hurry to sell just yet.
Fenerbahce would be able to offer Ziyech first team and Champions League football next season after finishing the league campaign as runner-up.
The Istanbul giants have made signing a winger a priority after parting ways with Luis Nani and Lazar Markovic.