"Disappointing to start this year with this shoulder injury. Will know the full extent on Monday," the 30-year-old former England captain said on Twitter.
"I'm seeing a specialist and will know more after that. It's all up in the air at the moment and it's hard to know where I stand."
Robshaw suffered the setback at Worcester last weekend.
Coach Eddie Jones faces a number of injury woes ahead of their first Six Nations game against France on Feb 4.
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"Chris will see a specialist early next week to determine the best course of action for his recovery; either an extended period of rest and rehabilitation, or an operation," the club said in a statement.
Billy and Mako Vunipola have been sidelined with knee injuries, while Joe Launchbury has a calf problem. James Haskell and George Kruis are doubtful with a foot injury and fractured cheekbone respectively.
Captain Dylan Hartley also faces a fitness test following a six-week suspension.
(Reporting by Pritha Sarkar, editing by Toby Davis)
