Spurs are thought to be leading the race for the highly-rated right-back, whose efforts helped Nottingham Forest return to the Premier League last season.
Spence amassed three goals and five assists from 49 appearances in all competitions for Steve Cooper's side, who also reached the quarter-finals of the FA Cup.
Interest in Spence has been rife since the England Under-21 international's breakthrough campaign, and he only has one year left to run on his Middlesbrough contract.
Arsenal were linked with a move for Spence before Tottenham moved to the head of the queue, and Lilywhites chairman Daniel Levy is understood to have been spearheading a second round of talks with Middlesbrough.
However, Sky Sports News reports that the two clubs are still apart in their valuations of Spence, with Tottenham so far refusing to meet Middlesbrough's £15m asking price.
The report adds that Tottenham are ready to move on if an agreement cannot be struck, and director of football Fabio Paratici is not involved in negotiations as he focuses on other targets.
Spence is reportedly open to a move to North London if the two clubs can ultimately agree a fee, but Tottenham are already said to have identified alternative targets for the right-wing back area.
Spence arrived at Middlesbrough from Fulham's academy in the summer of 2018, and the 21-year-old has two goals to boast from 70 appearances for the Championship club in all competitions.
Boro manager Chris Wilder recently insisted that his side will ask for a "fair price" for Spence, who is pining for the chance to play in the Premier League next term.
"Well we want the going rate. The negotiators, with my input, we've set a price and a fair price. We don't want to hold anybody back," Wilder told BBC Radio Tees Sport.
"Djed has made it clear that he wants to go play in the Premier League, he's earned that opportunity which he has done. You cannot get away from the fact that he's had an incredible season and with that comes interest but it has to be at that right price.
"While we'll not pay over the odds, we as a football club have got to set a fair and reasonable price for that to happen. I'm confident all three parties will get round the table and get this sorted out. If that happens then I'm sure we'll all walk away happy with the deal.
"We've got good players underneath, as we've talked about in terms of Izzy Jones and his progression last season. We've got a player who has had a fabulous season and got his rewards for that in terms of a big move and we've got the going rate for the England U21 top right wing-back of English descent."
Ivan Perisic, Yves Bissouma and Fraser Forster have already joined Antonio Conte's ranks this summer, and the Lilywhites boss is also keen to bring in a new attacker and centre-back before the window shuts. body check tags ::