Spurs chairman Daniel Levy has been spearheading negotiations with Chris Wilder's side for the 21-year-old while Fabio Paratici focuses on targets elsewhere.
Tottenham initially struggled to reach an agreement for the transfer of Spence, who is said to be valued at £20m by the Championship club.
Arsenal were also credited with an interest in Spence, who helped Nottingham Forest knock the 14-time FA Cup champions out of the competition last term.
Spence also earned promotion to the Premier League with the Tricky Trees, registering a total of three goals and five assists in 46 appearances across all competitions under Steve Cooper.
According to The Athletic, Tottenham have now made a breakthrough in negotiations with Boro, and a move is 'on the cusp' of being completed.
The report adds that Levy has maintained a strong relationship with Middlesbrough chairman Steve Gibson and Spence's agent Scott Smith, which has been a key factor in getting a deal done.
The England Under-21 international could now complete his medical at Hotspur Way before the end of the week, and Spurs will fork out an initial £12.5m to sign him from the Riverside Stadium.
The fee could potentially rise to as much as £20m with performance-related add-ons, and Boro boss Wilder recently admitted that he hoped a move would go through without any major roadblocks.
Speaking to the Northern Echo, Wilder said: "If nothing happens with Djed, then he's our player and he'll be back for pre-season. From my point of view, negotiations between Kieran (Scott), Neil (Bausor), Steve (Gibson) and Tottenham stay between Kieran, Neil, Steve and Tottenham.
"If the deal doesn't get done, we'll see where we go from there. Have I got an input into it? Yeah, 100 per cent. The boy obviously doesn't want to come back here, which I've got absolutely no issue with. He obviously sees his future away from here, and he had a brilliant season last season.
"We're expecting that deal to happen, but if it doesn't, then he's our player. From a club point of view, the deal has to be right. We're not trying to stand in the way of anybody. I've never done that as a manager, and I've had plenty of situations before where opportunities have come along to players.
"But there's always three parties involved in any deal, and it's got to be right for all three parties for it to happen. It's got to be right for the player, and obviously from the noises, it sounds as though that's (Spurs) where he wants to go. Tottenham want him, and we're prepared for that. But it's got to be right for us too. It all needs to marry up."
Spence joined Middlesbrough from Fulham's academy in the summer of 2018 and prepares to leave the club after scoring two goals in 70 appearances across all tournaments.
The Englishman would become Antonio Conte's sixth signing during a busy summer window, following Richarlison, Ivan Perisic, Clement Lenglet, Yves Bissouma and Fraser Forster to North London.
The Coupang Play Series clash with Sevilla on Saturday will come too soon for Spence to make his debut, but he could link up with the squad in time to face Rangers at Ibrox on July 23.