Lloris has spent the first half of the season out of the Spurs squad with Guglielmo Vicario and Fraser Forster named as Ange Postecoglou's first and second choice 'keepers.
And according to the Daily Mail, he is closing in on an exit to the United States with talks with LAFC said to be advancing.
Lloris has spent 11 years at Tottenham since joining the club from Lyon for an £11million fee in 2012.
The Frenchman - who won the World Cup in 2018 - has made 444 appearances in total for the club and was named captain in 2015.
However, he has not featured in a match since last season with his last outing coming in Spurs' 6-1 thrashing at Newcastle in April.
Lloris had been expected to leave Spurs over the summer but never managed to secure an exit and has since continued to train with the first-team squad.
When it became clearer he was not set to be in Postecoglou's plans, he lost the captaincy to Son Heung-Min.
