Kane twisted his ankle after scoring in the 1-0 win over Sunderland on Sunday and had to be carried off on a stretcher towards the end of the contest.
The 23-year-old was later seen leaving White Hart Lane on crutches and wearing a protective boot on his right foot, with manager Mauricio Pochettino conceding his forward may have suffered ligament damage.
Kane will undergo tests on Monday, but any spell on the sidelines will represent a major blow for Tottenham, who will turn to summer signing Vincent Janssen to fill the void.
"We just hope it is not too bad," Lloris said
"It is true that, if you look at the images, we can be a bit worried, but all the medical staff are going to try to bring him back as quick as possible.
"We have a competitive team and are fully confident in every player
When one player is missing, it gives an opportunity to another player to bring his skills, energy and quality.
"This is the story of the season and that's why we need all the players involved to be committed to the club because when the manager needs you, you need to respond well."
Kane was one of three Tottenham players to depart due to injury, after Eric Dier and Mousa Dembele had already gone off with cramp in their hamstrings.
Dembele, Dier and Kane were all key players in their side's draining title challenge last season and the trio were also involved for Belgium and England respectively at Euro 2016
Lloris sustained a hamstring injury last month after playing all seven of France's matches en route to the final, and the goalkeeper said more recovery time was needed.
"Of course we have a lot of players involved (at the Euros) at Tottenham, but it's the same case at some other teams
I think the medical staff will try their best," Lloris said.
"We all enjoy the way we work, but last season was a very, very tough season and very long, and you need time to recover that.
"But in football you don't have time because the new season is starting and you need to be ready to help your team and ready to compete."
Kane's injury was a blemish on an otherwise successful afternoon for Tottenham, whose victory over Sunderland sent them up to third in the Premier League and maintained their unbeaten start.
Pochettino had accused his players of a lack of passion following their Champions League defeat to Monaco last week, but Spurs responded with a dominant performance against the struggling Black Cats.
"Did the criticism hurt? Yes, but you know it is only the truth," Lloris said.
"He doesn't lie to his players and it was the case after the Monaco game
In the changing room after the game, it was a bit sad and we were very disappointed.
"We all felt guilty, but the most important thing is the way you react and today was a very good signal for the manager, the club and the fans too."
Lloris also hailed an "amazing" performance from opposite number Jordan Pickford, who kept his team in the match with a number of saves in the first half.
Pickford was finally undone when Kane capitalised on Papy Djilobodji's failed clearance, leaving Sunderland 19th in the table and with just one point from their opening five league games under David Moyes.
"We are a new back four and we're just getting to know each other still
We will get there," Pickford said.
"There have been changes
It starts with working hard in training, which we are doing, we are bonding together and the more we play together the better we will be
We will be OK.
"Tottenham are a top-quality side, that is why they finished where they did last season
We held our own
We could have been stronger in parts but we did OK."
Source : PA
Source: PA