Catalan publication Sport claims that the Blues are "willing" to pay €40m (£34m) for the young Spaniard, who has caught the eye with his performances at the Club World Cup.
Garcia has scored 3 goals in 4 appearances, including the winner against Juventus in the last 16 earlier this week.
Even so, Real may be open to selling the 21-year-old, but Sport suggest they may want to include a buyback option, which may put off the Blues from making an approach.
Why are Chelsea keen on purchasing the forward?
The London club are always on the lookout for the most talented young players and Garcia has certainly shown over the past few weeks that he possesses plenty of potential.
Another motivation for pursuing the Spanish Under-19 international could be doubts over the future of Nicolas Jackson.
Following his red card against Flamengo, there have been suggestions that the club are losing patience with his poor disciplinary record.
What is the latest on Jackson's future?
It is believed that Chelsea may consider offloading the 24-year-old if they receive an offer in the region of €55m-€60m (£47m-£51m), but they are not desperate to sell him. Speaking about the player's future, The Athletic's David Ornstein said:
"There is clearly some noise around the Nicolas Jackson situation, but I repeat, it's not something I know that Chelsea are actively trying to bring about."
"I think they've got huge respect for Nicolas Jackson, and what he has achieved so far, and the potential that he has for growth. I just know that he's the type of player and profile the number of clubs like. I've spoken to them. And whenever you mentioned Jackson, they say, he could become really, really good, and Chelsea will be aware of that themselves."
In other news, Chelsea are expected to sell a goalkeeper to a newly-promoted club for £21.5m.
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