The Lilywhites are believed to have identified the 21-year-old as their leading target for the right-back area following his stellar loan spell at Nottingham Forest last season.
Spence contributed three goals and five assists in 46 games for the Tricky Trees, who achieved promotion to the Premier League and qualified for the quarter-finals of the FA Cup in 2021-22.
Forest did not welcome Spence to the City Ground on a permanent deal, but he has been expected to leave Middlesbrough in search of Premier League football.
Tottenham have seemingly fended off interest from North London rivals Arsenal to move to the head of the queue for the England Under-21 international, with Daniel Levy spearheading negotiations.
A recent report claimed that Tottenham were edging closer to a £20m agreement for the transfer of Spence and were hopeful of wrapping up a deal for the defender before the weekend.
However, the Evening Standard now claims that Antonio Conte's side are in fact no closer to finding an agreement with Chris Wilder's side, who are standing firm on their £15m-£20m valuation.
Spurs are understood to have seen two previous bids of £10m and £11m rejected by Boro, who are yet to accept a bid from the Champions League contenders.
Boro boss Wilder recently expressed his hope that a transfer would go through without any major problems, as he understood Spence's longing to play in the English top flight.
"Hopefully, it will come to a quick conclusion because that would help everybody," The Northern Echo quotes Wilder as saying.
"That's the way we'd like it to be, although the situation isn't always straightforward. It is taking its time, but hopefully, it can get sorted. If not, then as I've said all along, Djed comes back to us."
Spence was signed by Middlesbrough from Fulham's academy in the summer of 2018, and the right-back has scored two goals in 70 appearances for the Riverside outfit in that time.
Tottenham struggled for consistency in the right-wing back area during the 2021-22 season, as Emerson Royal endured a hit-and-miss campaign and Matt Doherty suffered a season-ending injury back in April.
Emerson is also believed to have attracted the interest of Atletico Madrid, and Conte may be willing to part ways with the Brazilian if he can welcome Spence to the club in the coming weeks.
Richarlison, Yves Bissouma, Fraser Forster and Ivan Perisic represent a quartet of new faces at Hotspur Way this summer, and Clement Lenglet is on the verge of signing for the club on loan from Barcelona.
Tottenham will fly out for their pre-season tour of South Korea over the weekend, but a deal for Spence is not expected to be wrapped up in time for the defender to join the squad.