Coach Chris Fagan said that the former Essendon ruckman reported a sore foot before last week's pre-season camp at Maroochydore on the Queensland Sunshine Coast.
"He's had some scans on that and unfortunately he's got a stress reaction in his foot," Fagan told SEN.
"That's going to probably mean that he won't be available for selection until about the second week of the AFL season."
Draper only managed five games for the Bombers last year before a torn achilles tendon ended his season.
He and ex-West Coast coast captain Oscar Allen are the key additions for the Lions as they shoot for three-straight premierships.
Allen is expected to be available for their Opening Round match against the Western Bulldogs on March 7 at the Gabba.
The tall utility continues to recover from the knee injury that restricted him to 12 games last season.
Lincoln McCarthy and Keidean Coleman are also pushing for Opening Round returns after their seasons last year were ruined by injury.
Draper is the latest big-name player to be sidelined in pre-season - luckless Hawthorn midfielder Will Day (shoulder) and Adelaide defender Dan Curtin (knee) are also sidelined.
Carlton forward Jesse Motlop and North Melbourne defender Jackson Archer most likely will not play at all this year because of ACL knee ligament tears.
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