The 19-year-old only joined the Serie 'A' side from Barcelona last season, after netting 13 goals in 19 appearances for the club while on loan. His good form this campaign has led to interest from around Europe, and Chelsea and Spurs are thought to be watching him closely.
The Argentina-born striker now has an Italian passport, and the Daily Mail reports he would cost in the region of £2m.
Tottenham sent club scouts to watch the teenage forward on three separate occasions, including when he scored in the 3-1 derby day victory over local rivals Genoa.
The report claims Chelsea have also watched him in action, and are thought to be impressed by what they've seen.
Icardi joined Barcelona as a 15-year-old, but failed to make the grade at the famous La Masia academy, eventually joining Sampdoria on loan while the club were still in Serie 'B', although the young striker helped the Italian club to achieve promotion last season.
Chelsea have a young striker, Romelu Lukaku, currently out on loan at West Bromwich Albion, while Tottenham youngster Harry Kane is currently at Norwich City on a short term basis. But neither club has many options up front in terms of first-team strikers, and both are thought to be considering their options for January.