The 18-year-old has joined the Premier League side for a reported fee of £8.7 million, signing a five-and-a-half-year deal.
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Furo, a striker, has scored one goal in five league appearances since breaking into the first team, will go straight into Keith Andrews' squad.
Brentford boss Andrews said: "Kaye is someone we have tracked for a while that we are really excited about bringing to the club.
"His attributes are ones that I like and feel we can develop within the environment we have created. I have no doubt he will be a big player for us."