Chelsea, in particular, are said to be looking for a new forward in the summer transfer window as back up to Diego Costa.
The Blues currently have Loic Remy but he failed to establish himself last season, and Didier Drogba has left the club once again.
In contrast, Manchester City are thought to be looking to revamp their squad after a disappointing campaign last year and have scouted the striker.
However, both would face competition from Madrid as new manager Rafa Benitez looks to put his mark on his squad, and knows all about the forward following his spell with Napoli.
The Argentinian would fit the bill for all three clubs, having finished as joint top scorer in Serie A last season, scoring 22 goals for Inter Milan despite the Nerazzurri finishing eighth.
Icardi recently signed a new contract to tie him to the Italian side until 2019 but that has not detracted his suitors, and it is believed that a financially rewarding offer from any club could tempt Roberto Mancini’s side into a sale.