The 36-year-old has a little over 12 months left to run on his existing Spurs contract and has been the subject of criticism for a number of high-profile mistakes this season.
Lloris has a made a joint-high four errors directly leading to a goal - level with Manchester United's David de Gea - and he will also miss the rest of the season due to a hip injury.
Fraser Forster will continue to deputise for Lloris for Spurs' final three games of the campaign against Aston Villa, Brentford and Leeds United, and there is a consensus that the latter has now played his final game for the club.
Interim manager Ryan Mason refused to speculate over Lloris's future last week, but Spurs are understood to be in the market for a new number one this summer after 11 years with the Frenchman in between the sticks.
According to The Athletic, Lloris has recently been the subject of interest from an unidentified Saudi Pro League team, who are willing to treble his £100,000-a-week salary.
Should Tottenham sign a new goalkeeper this summer, the report adds that Lloris would not accept playing second fiddle and could consider a move elsewhere in order to continue playing regularly.
Nevertheless, the former France international is not thought to be in any hurry to relocate, and Tottenham would welcome him staying for the 2023-24 season given his experience and leadership skills.
Since leaving Lyon for Tottenham in a £10.9m deal in 2012, Lloris has turned out 447 times for the Lilywhites across all competitions, keeping 151 clean sheets in the process.
Lloris sits eighth in Spurs' all-time appearance list, and he is one of just 17 goalkeepers to have kept over 100 clean sheets, although he has failed to win a single major trophy in North London.
The World Cup winner was runner-up with Tottenham in the 2018-19 Champions League and 2020-21 EFL Cup, but he has just nine clean sheets to boast from 31 outings in all tournaments this term.
Amid increasing criticism of Lloris's performances, Spurs are believed to have earmarked Brentford's David Raya as an ideal replacement as they search for a shot-stopper more comfortable with the ball at his feet.
Raya has opened the door to a summer move elsewhere amid claims that he has rebuffed Brentford's attempts to tie him down to a new contract, and a fee of £35m could do the trick, but Spurs may face competition from Chelsea and Manchester United.
The likes of Valencia's Giorgi Mamardashvili, Aston Villa's Emiliano Martinez, Everton's Jordan Pickford and Leeds United's Illan Meslier have also been shortlisted, although the latter may have harmed his prospects of a move to a European-chasing club with a spate of poor displays recently.
Tottenham sit sixth in the Premier League table ahead of Saturday's trip to Villa, who are only three points worse off than Mason's side in the race for continental football.