The Magpies are still to make their first summer signing as pre-season gets under way.
This comes after reports last week, by the likes of the Daily Mail, that the initial bid, believed to be worth around £10.7 million, had been rejected.
Marseille have let a number of key players go in recent weeks with Andre Ayew and Dmitri Payet making Premier League switches, Gianelli Imbula heading to Italy and Andre Pierre Gignac signing with Mexican club Tigres.
Whether they will be willing to sell another of their top talents remains to be seen but the Newcastle offer is believed to have tempted the club president after Thauvin was signed for just over £7 million in 2013.
The flying winger was reportedly offered to Spurs earlier in the window, according to the likes of Metro, with Labrune hoping to start a bidding war, but the White Hart Lane club are said to have declined the advances.
Although the Magpies' current offer isn’t too wide the mark, the club’s recent sales have meant Labrune isn’t under pressure to sell and will likely be determined to squeeze a higher fee for the 22-year old.
The path now looks clear for Steve McClaren to get his man, with no other clubs seriously linked, but it is unclear whether owner Mike Ashley will stump up the extra cash needed to push the deal over the line.
Business has been slow to get started this summer at St James’ but the signing of Florian Thauvin would be a great opening gambit, and Newcastle fans will be hoping Ashley digs a little bit deeper in the coming days.