Spurs manager Mauricio Pochettino will not choose the defender's replacement until the end of the transfer window, but that will not stop the speculation about who it will be.
Belgium defender Jan Vertonghen's name has been mentioned as has Younes Kaboul, who has skippered Tottenham in both Premier League games under the former Argentina international.
Lloris, who recently penned a new five-year deal, would seem the most logical choice given his experience captaining the French national team, but it is not a role the goalkeeper feels he need to help Spurs prosper.
"I don't really need to wear the armband to feel like a leader for the team," Lloris, who skippered the side in the Europa League play-off first leg at AEL Limassol last week, said.
"I am very proud to the captain for the French national team. For Tottenham, it is just the manager's decision.
"I am not expecting something about that, I am just trying to do my job in the best way and I try to help the team to have the best performance.
"And after, in the changing room, I know that I have my place, and I try to be an example for everyone and especially for my young team-mates because in the squad there are a lot of young players."