The French star has been linked with Arsenal over the course of the past few months after a series of impressive Ligue 1 showings.
However, the Gunners are yet to make a serious move for the 23-year-old, which has tempted Newcastle into action.
The Magpies are keen to bolster their midfield ranks, and could up their efforts to tie up an agreement before next week’s deadline.
Alan Pardew already has a strong French contingent in his squad, and sees Grenier as an ideal addition as he plots to keep his side in the battle for European qualification.
As well as Arsenal and Newcastle, Liverpool have also been linked with the promising midfielder.
Meanwhile, the North East side are expected to tie up a deal for Remy Cabella.
The Monpellier playmaker has been on their radar for some time, with speculation suggesting a deal this month could be on the cards.
However, a swoop in six months’ time is more likely, at which time he may replace Yohan Cabaye.