The 37-year-old actor, who won the title of Sexiest Man Alive last year, has apparently told his hairstylist keep a lid on rumours about his luscious locks, reported Showbizspy.
"He brought in a special stylist while he films 'The Hangover' Part III and is paying boucoup bucks for her to sign a confidentiality clause," a source said
"He comes from a long line of bald guys and has used Propecia for year.
Cooper's romantic comedy-drama film "Silver Linings Playbook" has attained critical acclaim.
"Brad has thrown himself into work and is desperate for an Oscar nod in February, but he is paranoid about going bald. He thinks it could kill his leading-man status.
"He is been getting a cutting-edge hair-loss treatment and taking a battery of natural supplements