This is according to Spanish newspaper Sport, who understand Mourinho is considering signing Song as a back-up option for Nemanja Matic and Cesc Fabregas, a former teammate of the midfielder currently on loan at West Ham.
The Hammers have reportedly decided against signing Song on a permanent transfer at the end of the season, but the player is keen to stay in London beyond this summer after ruling out a return to the Camp Nou - and Chelsea would surely be an attractive option.
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Song's form has dipped in recent weeks
Song, who rose to fame with Arsenal before securing a dream move to Barcelona in 2012, struggled to cement his status as a first-team regular at the Camp Nou and was subsequently granted permission to leave the club on loan by Luis Enrique last summer.
The former Gunners’ star impressed in the opening months of the season at Upton Park, but his performances have taken a nose-dive over recent weeks, prompting West Ham to pull the plug on a permanent move.
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The cost of signing Song
In order to sign him, West Ham would need to pay Barcelona’s €10 million asking price, the player’s £80,000-a-week wage demands, plus signing-on and agents fees - and it seems the club’s owners cannot justify such a huge outlay on a midfielder who simply isn’t performing at this moment in time.
Despite his recent struggles, though, Mourinho may feel Song would be an upgrade on John Obi Mikel, who has made 13 Premier League appearances this season.
Song has plenty of big-game experience under his belt and, like Fabregas, would have learnt an awful lot during his spells with both Arsenal and Barcelona.
Chelsea fans unlikely to be impressed
However, news that Mourinho is keen on Song is unlikely to fill every Chelsea fan with joy.
That Song struggled so badly at Barcelona and is now failing to perform with West Ham will be a concern, while it could be argued that the African midfielder would not be an upgrade on Mikel, who may lack the star quality of his teammates but is certainly a useful back-up option.
Furthermore, Chelsea have several talented youngsters in their ranks, and Song’s potential arrival would knock them all back down the pecking order and leave them facing uncertain futures at Stamford Bridge.
The talented Ruben Loftus-Cheek, for example, is one midfielder with a very bright future ahead of him - but Chelsea fans may see little of him next season if Mourinho pays the €10 million to bring Song to west London this summer.
Chelsea fans: would you be happy if Alex Song signed for your club this summer? Tell us by leaving a comment in the box below!