The Parkhead club are looking at a number of centre-back options after missing out on number one target, Preston North End defender Ben Davies, to Liverpool.
According to Sky Sports, Celtic are keen on a loan move for Tanganga but face competition from two English Premier League clubs.
The 21-year-old England youth international has made six first team appearances for Spurs and is under contract with the London club until 2025.
Celtic are also hoping to land Everton right-back Jonjoe Kenny on loan until the end of the season.
The former England Under 21 defender has attracted interest from Burnley but is expected to join the Scottish champions.
Kenny, who spent last year on loan at Schalke, has 30 Premier League appearance to his name but has only featured once in the top flight this season under Carlo Ancelotti.