Lloris, 32, suffered the injury during a 3-0 defeat to Brighton & Hove Albion on October 5 and is expected to miss the remainder of 2019.
"Everything's well. There can always be hazards, but my goal is to be ready for January. Whether it's the beginning, the middle or the end, I don't know. In any case, I'll take time to make sure I'm well healed and that I can come back in top form.
"I'm also using the time to regenerate myself. I've never been away from the pitch for so long, so it's something new, but I'm treating it positively. I'm using this period to spend time with my family and my children.
"I'm in good hands and we're moving forward calmly. Everything's been done as it should have been. Now, we have to focus on rehabilitation, strengthening, gradually gaining mobility and confidence."
Tottenham recently confirmed Lloris had undergone surgery on his left elbow after it was found to still be unstable one month after the injury occurred.
Lloris captained France to victory in the 2018 FIFA World Cup before he led Spurs to the final of last season's UEFA Champions League, where they finished as runners-up to Liverpool.
The stopper received a nomination for the 2019 Ballon d'Or on the back of those achievements, though his chances don't look positive following Tottenham's horror start to this season:
Lloris kept one clean sheet in nine appearances prior to his injury, a 4-0 win over Crystal Palace in north London.
Replacement Paulo Gazzaniga has three clean sheets in nine games across all competitions: Home and away to Red Star Belgrade in the Champions League, as well as a 0-0 draw against Colchester United in the Carabao Cup third round (Spurs lost 4-3 on penalties).
Tottenham podcaster Ricky Sacks recently appeared on Love Sport Radio and said he was unsure as to whether Gazzaniga was up to the task of becoming Mauricio Pochettino's recognised No. 1:
Love Sport Radio @lovesportradio
🗣️ "I'm amazed he's still at Tottenham" @RickSpur from @LastWordOnSpurs thinks that although Paulo Gazzaniga has looked excellent he's unsure whether he's good enough to be Tottenham's permanent number one. #THFC 📻 Tune in - DAB 📲 Listen - https://t.co/XWhRQh9myF https://t.co/bf4IsZDVfD
Those remarks could bode well for Lloris, whose contract at Tottenham Hotspur Stadium isn't due to expire until the summer of 2022.
The Champions League pool stage will end on December 11, but Pochettino's side will clinch at least second place in Group B if they beat Olympiakos at home on November 26.
Lloris will hope his side remain in Champions League contention by the time he's fit to return, though Pochettino will be wary of rushing his first-choice stopper back from serious injury.