Murphy is believed to be one of Rafa Benitez’s key transfer targets as he looks to improve his squad in the wide areas.
Newcastle’s bid is believed to be somewhere in the region of £10 million, which will include a host of performance-based add-ons.
The Norwich youngster is believed to be keen on making the move happen this summer and has reportedly informed his agent of his desire to join the club he supported as a child.
Norwich City sporting director Stuart Webber has confirmed that the Championship side will sell the player, but only if the deal is right for the club.
The Magpies have already signed Christian Atsu on a permanent deal this summer but will need to continue to spend as they prepare for life back in the Premier League.