In Levy's eyes, Spurs are one of the Premier League's so-called "Big Six," with a stadium and training ground on a par with, or better, than any of their domestic and European rivals. They are a club with a proud history, a global fanbase and -- this is important to overseas players and coaches -- a team that plays in London, one of the world's great cities.
But while the game's top coaches will unquestionably share Levy's view of the stadium and training ground, they also see a team in need of a rebuild and a regime in charge which has consistently been reluctant to spend the sums required to make Spurs truly competitive.
They see a club that have won just one trophy this century, the EFL Cup in 2008, and have not brought home the league title since 1961. And, perhaps most importantly, they realise that their first task as manager is likely to be finding a replacement for their best player, centre-forward Harry Kane, who has made it clear that he wants to move to another club this summer.
Since Mourinho was sacked less than a week before the Carabao Cup final against Manchester City in April, and replaced by the 29-year-old interim coach Ryan Mason, Spurs have seen an array of top coaches either find other jobs or reject the opportunity to move to the Tottenham Hotspur Stadium.
Massimiliano Allegri has returned to Juventus, Carlo Ancelotti has taken charge of Real Madrid for a second time, Julien Nagelsmann has moved from RB Leipzig to Bayern Munich and Hansi Flick, who guided Bayern to the Champions League title in 2020, is expected to replace Joachim Low as Germany coach after Euro 2020.
Former Spurs manager Mauricio Pochettino, now at Paris Saint-Germain, has committed himself to the French club after sources told ESPN that he was in talks over a return to Tottenham last month, while former Chelsea boss Antonio Conte, having resigned from Inter Milan after winning Serie A this season, pulled out of talks last week due to his expectations being out of sync with the club's.
The pursuit of Conte was in keeping with Tottenham's chaotic pursuit of a new manager. The Italian is a coach with a winning pedigree, but he also demands big money for top players and his abrasive approach has upset his bosses and players at previous clubs. In many ways, Conte has an even more challenging personality than Mourinho, but Spurs pursued him regardless, despite the way it ended for the previous manager. Replacing one demanding coach with another is probably not the best way to go for Tottenham right now.
On May 19, Levy signed off an open letter to the Spurs fans in which he outlined the criteria for a new Head Coach.
"We are acutely aware of the need to select someone whose values reflect those of our great club and return to playing football with the style for which we are known," he wrote. "Free-flowing, attacking and entertaining -- whilst continuing to embrace our desire to see young players flourish from our Academy alongside experienced talent."
Read that again and try to find anything which fits with the pursuit of Conte.
Officially, the search for a new Spurs manager is being led by Steve Hitchen, the club's technical performance director. Sources have told ESPN that agents representing coaches have been informed that Hitchen is looking for a manager with a long-term vision to build the team from the bottom up, promoting the club's promising young players and, in many ways, rewinding the clock to the start of the Pochettino era.
There is a preference for a British coach, or at least a coach with Premier League experience, which is why there is support internally for Brighton's Graham Potter. Belgium's Roberto Martinez, who has managed in the Premier League with Wigan and Everton, is another name under consideration.
But despite the Hitchen blueprint to take a long-term approach with a less glamorous candidate than Mourinho, Conte or Pochettino, Spurs continue to make advances for the big names.
There is a valid reason for this. With season tickets to sell and a stadium to pay for, sources said that the Spurs hierarchy believe that an established coach with a winning track record is more likely to persuade supporters to dig deep and pay to fill the stadium than a coach with a less impressive pedigree.
But the problem for Spurs is that none of the A-list coaches are interested. Why risk a hard-earned reputation by taking charge of a club that will start next season in the Europa Conference League?
There are still some unemployed big-name coaches out there, such as Zinedine Zidane and Maurizio Sarri, but even if the job appealed to them, there is no guarantee that either would be a good fit.
At some point, and it needs to be soon, Spurs need to decide exactly what they want. The club hierarchy want a big name and the football staff believe that a team-builder is the only way forward, but Levy will be the one to ultimately decide. He just needs to find somebody, anybody, that shares his vision of Tottenham Hotspur.