On Wednesday, the French players' union (UNFP) announced its nominees for the 2021-22 awards, with the pair missing out alongside head coach Mauricio Pochettino.
The Argentine coach is facing an uncertain future at the Parc des Princes, sources told ESPN last week, with an expectation that he will leave this summer.
PSG have lost just four times this season as they claimed the Ligue 1 title last month, the first of Pochettino's coaching career, but it was largely overshadowed by a Champions League round-of-16 exit to Real Madrid.
Messi has suffered a lacklustre maiden season in France, scoring four league goals in 23 games and registering 13 assists, while Neymar has scored 11 goals in Ligue 1.
The pair have been jeered by the club's fans in recent months, and Neymar took a swipe at his own supporters for walking out of the Parc des Princes as they drew 1-1 to Lens last month to clinch the title.
"It was surreal that part of the crowd left the stands," Neymar told ESPN Argentina. "They're going to get tired of whistling, because I have another three years left on my contract."