The former Aston Villa playmaker has been on trial at the Stadium of Light since the beginning of the summer.
N'Zogbia is set to be given an extended run-out against Montpellier
The former Newcastle United midfielder has been on trial with the Black Cats since the first day of pre-season, featuring in friendly wins over Hartlepool United, Rotherham United, Stade Nyonnais and Dijon.
He is yet to be named in a starting line-up by Moyes, who is still contemplating whether to make the 30-year-old the club's first signing of the summer.
However, the South Shields Gazette claims the Frenchman will get an extended run-out in this weekend's friendly, with Moyes understood to be eager to make a decision about whether to make a move.
N'Zogbia is a free agent after seeing out a £50,000-a-week contract at Aston Villa last season, which saw him make just two appearances during a disastrous campaign that eventually saw them relegated.
It is not the first time he has been touted for a move to Wearside. Former boss Steve Bruce made an unsuccessful attempt for his signature prior to his move to Villa Park in the summer of 2011.
N'Zogbia is understood to be keen on striking a permanent deal, and believes he has done enough to earn himself a contract at the Stadium of Light.