The 24-year-old summer signing injured himself in training on Friday as he went to take a shot and will see a consultant this week.
He joins Ryan Brunt and left-back Gary Sawyer as long-term absentees.
"It's a huge blow, any players we have in the squad that are injured is very disappointing for us," manager Derek Adams told BBC Radio Devon.
Spencer had become first choice in the central striking role of the League Two leaders following an injury to on-loan Bristol City forward Paul-Arnold Garita, who has just returned to fitness.
"He and Garita have played in that area this season and held the ball up and performed well for us," Adams added following Saturday's 3-0 home loss to Grimsby.
"It's hard to take as a manager because you feel that you're getting somewhere and then something happens, it's like a roadblock, it's very difficult to overcome these things, so it's hard to take."
Plymouth Argyle striker Jimmy Spencer will miss at least three months after breaking and dislocating his ankle.