Fontaine has been playing through the pain barrier in recent matches.
He could face Burnley at Ashton Gate on Saturday, but it will be the player`s decision.
"It just depends on how he is feeling," Millen told the club`s official website. "If he says it is too sore then he won`t play and we will press on with an operation on Monday.
"If he has an operation then that will be his season over."
Source: PA
Source: PA