Mbappe under fire for skipping France duty while fit enough to play for Real Madrid
The forward picked up a muscle injury last month but made his return to the pitch off the bench in his club's Champions League defeat at Lille last Wednesday.
France coach Didier Deschamps left Mbappe out of the squad he announced on Thursday, saying he lacked preparation, having played too few matches.
Mbappe started Real's LaLiga win over Villarreal on Saturday.
"Either you're injured, and you don't play with your club and you're not called up in the national team," former France international Maxime Bossis told French sports daily L'Equipe on Monday.
"But when you come off the bench in the Champions League and you start a league game, it makes things fuzzy. He's a special player. We've known that with Michel Platini, when we still wanted him with us when he was injured."
In a video released by the French Football Federation (FFF) on Monday, Deschamps said: "Kylian embodies a lot of things... The interests of the clubs and the interests of the national teams inevitably diverge.
"We must also remember that the employer is the club, not the federation."
Mbappe's situation is more complicated because the former Paris St Germain player is the France captain after being handed the armband ahead of Antoine Griezmann, who last week announced his retirement from international football.
France, second in Group A, face Israel in Budapest on Thursday and travel to Belgium next Monday.
"(The captaincy) is on my mind. But, as I do every time, the key is to discuss it with the players involved," coach Deschamps added.
"Obviously, with Kylian our captain, and Antoine, who was the vice-captain, both missing, it means there will definitely be others for these two matches."
Griezmann's commitment to Les Bleus was never in doubt as the Atletico Madrid forward played a record 84 consecutive matches for his country between 2017 and 2024.
"As a captain (Mbappe) must be an example for the fans, and he has not been," Fabien Bonnet, spokesperson for the Irresistibles Francais, the main France supporters' group, told L'Equipe.
"What emerges is that the real captain, Antoine Griezmann, is gone."
On choosing the skipper for the Nations League games, Deschamps said: "It’s more about if (I can find) certain players who can handle this responsibility in the moment without it affecting their behaviour or performance.
"That’s why it’s important for me to really know the players on a personal level - their character, their sensitivity -relative to the group."
