Boro are asking for a fee of £15million up front rising to £20million with add-ons, having held firm in negotiations in previous weeks.
The 21-year-old is hot property this summer after an impressive campaign with Nottingham Forest last term.
During his promotion-winning season on loan at the City Ground, Spence made 46 appearances for the club, scoring three goals and creating an additional five.
An attack-minded wing-back by nature, the England under-21 international looks set to become a new addition for Antonio Conte's side as he looks to build a team capable of challenging in the future.
Should the deal go through as expected, Spence would be Spurs' fifth adddition of the summer following deals for Yves Bissouma, Ivan Perisic, Fraser Forster and most recently, Richarlison who arrived from Everton for a fee of £60million.
The wing-back is undoubtedly a top-level talent, however a future at Middlesbrough always seemed unlikely following a fallout with then-manager Neil Warnock, leading to his loan spell with Nottingham Forest.
However the 73-year-old has said that he's pleased to see Spence getting the recognition that he deserves.
Speaking on talkSPORT Breakfast, his former Boro boss said: "I had a few run-ins with Djed, to be honest, as managers do.
"I had one or two chats with him and a few words were said. So the lads said, he had a bit of a go at me.
"But I'm glad to see it seems the penny has dropped because, like I said to him face to face, it could go either way.
"I told him, 'you could be playing for Bromley next year, or you could be in the Premier League'.
"It looks like he's going to be in the Premier League, so good luck to him."