Mbappe, 24, had been touted as one of the candidates for the role since goalkeeper Lloris retired from international football in January.
A goal in the 11th minute of stoppage-time by Antoine Griezmann earned Atletico Madrid a 2-1 home win over 10-man Porto in the Champions League on Wednesday.
There were early suggestions surgery could see the World Cup winner miss Qatar 2022, but he has opted instead for therapy.
Atletico Madrid conceded a last-minute penalty kick as they were held to a 2-2 draw by struggling Levante in LaLiga on Thursday.
Messi received a straight red card for the first time in his career.
