The Magpies have endured a largely frustrating transfer window, with French defender Florian Lejeune and Ghanaian forward Christian Atsu the only arrivals.
But the north-east club now look set to add 22-year-old Murphy to their ranks after agreeing a fee in the region of £12 million with the Championship side.
Newcastle are believed to have had an £8 million bid rejected last week for England under-21 international Murphy, who blossomed in the second tier last season, scoring 10 goals in 40 appearances for Norwich in all competitions.
Murphy is set to undergo a medical and discuss personal terms with Rafael Benitez's side.