The Englishman claimed gold at London 2012 as part of an excellent amateur career, but this was by far the toughest test since he turned professional.
Campbell endured a bad start, being floored in the second, and while he recovered in the later rounds, the Venezuelan used all his experience to close out the win.
Scot Victor Loughlin scored the contest 115-113 to Campbell, however the other judges gave it to Linares with scores of 115-112 and 114-113.
The defeat is the second of Campbell's career after his shock loss to Yvan Mendy in 2015, while Linares - whose last two fights were points victories over Mancunian Anthony Crolla - extended his record to 43 wins.
The first round was a cagey affair with Linares landing a good right hand and it was the right hand that put Campbell to the canvas in the second.
A nasty cut opened under Campbell's right eye and while the Yorkshireman recovered well, Linares was still landing the better shots and Campbell was left with too much to do in the second half of the contest.
He was excellent in the middle and later rounds and fought his way back to make it close on the scorecards, but Linares never looked particularly troubled.
In the ring afterwards, Campbell said: "No-one can ever doubt my heart.
"I got off to a rocky start, he hit me with a nice clean shot in the second round, caught me on the eye and cut it. I wasn't dazed or anything like that, just a nice shot that put me on my back.
"From then I had double-vision in one eye for the rest of the fight."
He added: "I thought I won the fight. He's a great champion but I thought I outclassed him. I didn't think he was landing any shots whatsoever and I was catching him with all the clean shots."
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