Fabio Paratici, Spurs' incoming sporting director, reportedly narrowed down the club's managerial search to Fonseca, who left Roma after two seasons in charge. Paratici served as Juventus' general manager during Fonseca's time in Italy.
Tottenham reportedly attempted to hire Antonio Conte last week, only for negotiations to break down over salary and recruitments.
If hired, Fonseca would effectively swap positions with Jose Mourinho, who took the job in Rome.
Mourinho was fired as Spurs manager in April.
Fonseca's coaching career took off in earnest at Shakhtar Donetsk. He won three consecutive domestic doubles between 2017 and 2019, earning widespread praise for his energetic pressing and counterattacking play.
Hoping to achieve a similar style of play, Roma hired the Portuguese tactician in June 2019. But the 48-year-old largely struggled to implement his philosophy, and the club ultimately failed to qualify for the Champions League in both of his seasons as manager. His greatest success came in the Europa League, with the Giallorossi making a run to the semifinals. Manchester United beat them 8-5 on aggregate.