Spurs currently lead the queue of clubs interested in the 21-year-old but the first round of talks resulted in no agreement.
Boro value the right-back in excess of £15m plus add-ons, which created an impasse during negotiations with Spurs unwilling to match that fee.
It has emerged that Tottenham’s pursuit of Spence is being led personally by Levy, leaving football managing director Fabio Paratici to focus on other targets.
The defender made a total of 50 appearances for both Middlesbrough and Nottingham Forest in all competitions during the 2021/22 season, scoring three times and setting up a further five goals.
The 21-year-old achieved promotion to the Premier League with Forest via a playoff win over Huddersfield Town, but has since returned to Boro at the end of the loan deal.
Spence is also keen to join Antonio Conte’s side for the next season, who are readying themselves for another approach for the wing-back.
Spurs qualified for the Champions League next season by finishing in fourth place ahead of north London rivals Arsenal at the end of the 2021/22 season.