The Welshman has been left unhappy by the departure of Rafa Benitez, who was axed to make way for club legend Zinedine Zidane.
However, even though Bale would bring an element of superstardom to Stamford Bridge, the club should instead attempt to sign three different Madrid players.
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Here are the three individuals in question and the ones that could help Guus Hiddink bring a feel-good factor back to the home of the champions.
Okay, so obviously Alvaro Arbeloa is nowhere near in the same category as Bale, but the former Liverpool defender would provide good competition for Branislav Ivanovic at right-back.
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The Serbian international has been a complete contrast to his former self and has struggled at the back all season long.
His lack of pace was particularly exploited by the performance of Anthony Martial in the goalless draw with Manchester United last month, and Arbeloa would be a shrewd signing to help get him back to his best.
The Spaniard was part of the side that won La Liga under Jose Mourinho in 2012, and could be on the move with Madrid possessing Danilo and Dani Carvajal in his position.
Prior to the sudden departure of Benitez, Isco was at odds with his manager's tactical decisions and selection policy during his brief spell at the Santiago Bernabeu.
He has been regularly linked with a move to Manchester City due to his healthy relationship with coach Manuel Pellegrini, and the Chilean was lining up a huge bid to land his man.
However, the arrival of Zidane has thrown a spanner in the works, with the Frenchman now unlikely to listen to offers for the midfielder.
That would not necessarily deter Chelsea, with uncertainty continuing to linger over Oscar’s future given his training ground bust-up with Diego Costa.
Costa was Chelsea’s best player last season, blowing teams away with his strength and power, but the Spaniard has scored just six league goals so far this season.
Karim Benzema remains the sole reliable option at Madrid, with the club allowing Gonzalo Higuain and Alvaro Morata to leave in recent seasons.
Radamel Falcao and Loic Remy have proven unreliable when given a starting role at the club, and the lack of a good striker has been exposed in a tame year in front of goal.
Benzema could be the man to fire Chelsea back to the top, and could move if Zidane decides to play Cristiano Ronaldo as the club’s main striker.
Who should Chelsea sign in the summer? Have YOUR say in the comment box below!