Spurs are looking for a new manager after the sacking of Jose Mourinho in April, who was in charge for just 18 months.
Pochettino is understood to have told PSG chief Leonardo that he wishes to leave the Parc des Princes, although the French club are seemingly unwilling to let him leave after just six months at the helm and are set to trigger the one-year extension in his contract.
According to the Daily Mirror, Pochettino's first choice were Spanish giants Real Madrid, however Levy believes he can convince his former employee to return to north London this summer.
Tottenham's first choice target was former RB Leipzig manager Julian Nagelsmann, who has since agreed to coach Bayern Munich next season.
Ajax head coach Erik ten Hag is still believed to be on their shortlist, even though the Dutchman recently signed a new deal with the Eredivise outfit at the end of April.
Former Chelsea boss Antonio Conte, who left Serie A champions Inter Milan last month, is also believed to be on Tottenham's list of managerial targets.