The 29-year-old lost on a split decision to his Venezuelan opponent on Saturday night, having recovered from being knocked down in the second round.
Scot Victor Loughlin had Campbell winning 115-113, but the other two judges gave victory to Linares with scores of 115-112 and 114-113.
Following the bout in the Californian city of Inglewood, the Hull-born boxer admitted that he had been mourning the loss of his father Bernard, who was diagnosed with cancer in 2014.
"I probably cried once a day. I had to try and shut feelings off. After the fight I had a good cry," Campbell told BBC Sport.
"If someone had found out I would have denied it. I didn't want Linares' camp thinking it was a weakness. I didn't want them thinking I was hurt.
"The only thing that kept me going is I know what my dad would have wanted for me. To fight and to win."
The defeat to Linares is the second of Campbell's career after his shock loss to Yvan Mendy in 2015.