Spurs are reportedly looking to add up to six players this summer and have been linked with moves for free agent Paulo Dybala and Southampton captain James Ward-Prowse, but it is Saints goalkeeper Forster who looks set to be their first summer recruit on a free transfer.
Forster is set to leave Southampton when his contract expires next month and Conte has identified the 34-year-old as the ideal back-up to Hugo Lloris.
Club captain Lloris played every Premier League game last season as Spurs ended the season in fourth position, confirming their return to Champions League football.
But the club is seeking adequate back-up between the posts, with current No.2 Pierluigi Gollini expected to return to Atalanta following his loan spell in north London.
The move will see Forster bring the curtain down on his eight-year career at St Mary's. The Englishman joined Southampton in 2014 after a successful spell at Celtic, where he won three Scottish league titles and two Scottish Cups.
He returned to Glasgow on a season-long loan for the 2019/20 campaign, where he added a fourth Scottish Premiership title and Scottish League Cup glory.
Forster has been capped six times by England, making his debut in a friendly against Chile in 2013 under Roy Hodgson and selected in the England squad for the 2014 World Cup in Brazil.
His arrival at Tottenham would signal the first piece of transfer activity for Conte, ahead of what appears could be a busy summer ahead at the Tottenham Hotspur Stadium.
It's claimed Spurs have identified six new signings to strengthen their squad for next season's assault on the Premier League and Champions League.
The club have been linked with a move for Aston Villa's John McGinn, in addition to Saints skipper Ward-Prowse, as Conte looks to provide depth to his midfield.
Spurs are also expected to tie up permanent deals for Dejan Kulusevski and Cristian Romero who have both impressed since joining the club on loan in January.
Dybala has long been linked with a switch to north London with the Argentine out of contract at Juventus and free to move to a club of his choosing this summer.
Reports in Italy suggest Inter Milan are leading the chase for his signature, with Jose Mourinho's Roma also showing an interest in recent days.
Dybala has enquired about living in London ahead of a potential Premier League switch, but a stay in Serie A seems likely.