The Magpies, who are still searching for a replacement for Demba Ba after he moved to Chelsea earlier this month, are believed to have offered 750,000 euros (£628,000) to the Ligue 1 side for the 26-year-old.
However, Sud Ouest and L'Equipe both report Bordeaux have turned down the offer as they have put a price tag of one million euros on the former France Under-21 international.
Gouffran has scored 10 goals in 22 games for Francis Gillot's side to help them to seventh in the table.
Gouffran, who moved to Bordeaux from Caen in 2008, only has six months left on his contract with the French club.
It is believed Bordeaux want more for the player, but Newcastle are hopeful of striking a deal.
Arsenal boss Arsene Wenger is known to be keen and has been strongly linked with the former France Under-21 international.
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