The 21-year-old ended a brief spell at Manchester City last year to search for more game time at St Mary's, where he quickly became the Saints' number one goalkeeper.
Bazunu started 32 Premier League games on the bounce for Southampton, but he only kept four clean sheets and has been dropped for Alex McCarthy in the Saints' last three games.
In Bazunu's absence, Ruben Selles's side have suffered three successive losses while conceding eight goals, and they will be relegated to the Championship on Saturday if they fail to beat Fulham.
The Republic of Ireland international is contracted to Southampton for another four years, but he could be part of a mass exodus of players if the Saints' time in the Premier League comes to an end.
Despite Southampton's standing at the foot of the Premier League table, Football Insider claims that Tottenham have been left impressed with Bazunu's showings and are weighing up a move for the summer window.
The Lilywhites are reportedly set to focus on signing young talents with a high ceiling for the 2023-24 season and beyond, and Bazunu would fit the bill.
The 6ft 2in shot-stopper left Shamrock Rovers in 2019 to join the academy ranks at Man City, and he was also sent out on loan to Rochdale and Portsmouth during his time at the Etihad.
Bazunu ultimately departed without making a single senior appearance for Pep Guardiola's team, joining Southampton in a £12.3m deal, and the Saints could expect a profit on any future sale.
The 21-year-old has also earned 14 caps for the Ireland national team as he battles for the gloves with Liverpool's Caoimhin Kelleher, who is also thought to be admired by Tottenham.
The Lilywhites are being tipped to move on from current number one Hugo Lloris in the summer window amid criticism of the 36-year-old's performances, having made a joint-high four errors leading to a goal this season.
Lloris, who has been ruled out for the remainder of the campaign with a hip injury, is under contract in North London for another 13 months, but he has reportedly been offered a triple-your-money deal by an unnamed club in Saudi Arabia.
The former Lyon man, who has been Spurs' number one for the past 11 seasons, is not thought to be in any rush to leave the club, but at the same time, he is not willing to act as the club's number two next season.
Aston Villa's Emiliano Martinez and Brentford's David Raya have also been linked with moves to Tottenham, who have been warned by Thomas Frank that the latter will cost at least £40m this summer.