Forster is impressing on-loan at Celtic, which want to sign him permanently.
The Daily Express says the on-form Englishman is on the radar of Harry Redknapp, who sent his goalkeeping coach, Tony Parkes, up to watch him keep another clean sheet in last week's Old Firm win.
Spurs are looking for a long-term replacement for 40-year-old Brad Friedal and Forster, who has kept five clean sheets out of the last seven games, is amongst the keepers they are considering along with Siena's Zeljko Brkic.
Celtic manager Neil Lennon is aware of the interest and has urged club chief executive Peter Lawwell to tie up a permanent £2 million deal for Forster.
Tottenham would have to match that £2 million fee and also buy-out Celtic's option although any move would be well within their financial capabilities.