Mignolet's career on Merseyside has depreciated in recent years, and the stopper could be shown the door in the weeks ahead.
Alisson Becker is established at the Reds first-choice No. 1, and Mignolet needs a fresh challenge at a new club.
Belgian newspaper Het Belong van Limburg (h/t Adam Skinner of the Daily Express) reported Vieira is in the market for a new goalkeeper and could offer Mignolet the chance to play first-team football in Ligue 1. Current Nice stoppers Yoan Cardinale and Walter Benitez have reportedly not met Vieira's expectations since the former Arsenal legend took over the reins in June.
Mignolet's reputation slowly collapsed after arriving from Sunderland in 2013, and the player never reached the heights expected of him at Liverpool.
Manager Jurgen Klopp kept faith with Mignolet and fellow stopper Loris Karius until the end of last season but then broke the bank to capture Allison from Roma.
Liverpool paid £67 million for the Brazil international, and Mignolet has been left on the bench in a reserve capacity.
Mignolet would be perfectly suited for French football, and the player has remained in Belgium's squad despite a tough time in the Premier League.
The 30-year-old was present in the Red Devils squad for his country at the 2018 FIFA World Cup in Russia, will Belgium finishing the competition in third.
The Belgian remains a solid shot stopper, but his decision-making has let him down at vital moments during his spell at Anfield.
Liverpool fans have never quite trusted his overall judgement, and most were pleased when Klopp recognised the club needed fresh blood between the sticks.
Vieira is a young, progressive manager, and with Nice in mid-table, Mignolet could become a major building block of the coach's fresh approach in the south-east of France.