And they are also understood to be chasing Liverpool target Christian Benteke, who has a release clause at Aston Villa of around £32.5m.
Chelsea have just Diego Costa and Loic Remy as recognised first-team strikers and Mourinho has already begun talks with Radamel Falcao over a switch.
The former Manchester United loanee is set to be shipped out by Monaco again next term, with Mourinho hoping he can get more than the four Premier League goals Louis van Gaal got out of him.
Mourinho's hopes of bringing Liverpool's Raheem Sterling to Stamford Bridge have suffered a blow after Manchester United emerged as suitors of the player.
According to ESPN, United have made a £25m initial bid for the England winger.
Contract negotiations with Liverpool have broken down after Sterling rejected a £100,000-a-week deal in the spring.
And United believe they can pip Chelsea, Arsenal, Manchester City and Real Madrid to the ace, with Louis van Gaal ready to spend his £150m transfer kitty.
GETTY
Radamel Falcao is reportedly close to signing for Chelsea
The striker, 22, was valued at £32m and was thought to have caught the eye of the Blues and Arsenal.
But a statement on the Serie A club's website read: "FC Internazionale is pleased to announce that Mauro Icardi has inked a new contract with the club.
"The Argentine striker's new agreement runs until 30 June 2019."
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