Spurs were believed to be close to announcing Fonseca as their new manager, with the Portuguese emerging as the leading candidate to become Jose Mourinho's successor.
However, according to Sky Sports News, Tottenham have decided to end negotiations with the 48-year-old and are now looking at other options for the position.
The report claims that Gattuso is at the head of the queue, with the 43-year-old available following his departure from Fiorentina, where he spent just 23 days in charge.
The former AC Milan and Napoli manager is believed to have left La Viola following a disagreement over player recruitment during this summer's transfer window.
Gattuso, who started his coaching career with Sion in 2013 after retiring as a professional footballer, is said to have the full support of new Spurs managing director of football Fabio Paratici.