According to Italian outlet Cronache di Spogliatoio, Spurs are weighing up a move for the 33-year-old after failing to score in any of their opening four league matches.
Icardi left Galatasaray this summer as a free agent following a four-year stint in Turkey, having previously played for Inter Milan and Paris Saint-Germain, and remains without a club since the transfer window closed.
Argentine journalist Nahuel Ferreira has reported that Olympiacos made "formal contacts" over a deal last week, though nothing has been confirmed, with the Greek club reportedly wary of matching the roughly £8.6 million annual salary Icardi commanded in Turkey.
Tottenham's need for a fresh attacking option is clear. They have conceded eight goals in the league this season, sit 17th in the table with two defeats and two draws from four games, and were knocked out of the Carabao Cup by Liverpool on Tuesday having beaten Championship side Charlton Athletic 5-1 in the previous round.
That exit came despite a busy summer of spending that brought in Sandro Tonali for £100 million and Mateus Fernandes for £85 million, alongside Omar Marmoush on a season-long loan from Manchester City with a £60 million obligation to buy, and winger Savio for a further £85 million.
None of Spurs' forward options have yet found consistent form in front of goal. Marmoush, who started in the goalless draw with Liverpool having also featured against Newcastle and Nottingham Forest, showed more quality than in recent weeks but still could not find the net, while Dominic Solanke has failed to score in five appearances this season.
Richarlison, meanwhile, has not played since Tottenham's opening-day 3-0 defeat by Brentford and is expected to remain sidelined until at least January following a breakdown in his relationship with the club.
Whether Icardi represents a genuine solution remains to be seen, and a move is regarded as unlikely, but Tottenham's toothless start to the campaign means De Zerbi will welcome reinforcements of almost any description as he looks to arrest a troubling run of results.