The 33-year-old sustained the injury during Saturday's win over Wakefield Trinity, ruling him out of next week's Challenge Cup semi-final as well as the final if Rovers make it.
Linnett has been a key figure at Craven Park since joining the Robins ahead of the 2019 campaign, winning back-to-back player of the year awards and earning a place in last season's Super League Dream Team.
KR head coach Tony Smith is preparing for the worst as Linnett waits to discover the true extent of the injury.
“He tore his bicep and it's hanging on,” said Smith.
“He’s seeing a consultant tomorrow and it’s not looking great based on the scans.
“If he needs repair it’ll be a 12-week repair. It could be a blow and most probably will be a blow to us.”
The loss of Linnett could be offset by the return of fellow second-rowers Frankie Halton and Luis Johnson in time for the Elland Road date with Huddersfield Giants, while Matt Parcell is fit for this Friday's trip to Leeds Rhinos.
But prop Tom Garratt is still struggling with concussion symptoms after taking a knock to the head in KR's win over Warrington Wolves at the start of April.
“It’s a big concern for us about him,” said Smith.
“We’re doing everything that’s required in terms of making sure he’s seeing consultants and getting the absolute best care possible.
“He hasn’t quite been himself since. Whether that’s concussion-related or some other things that may be bothering him like a viral infection, we’re not totally positive.
“He’s having more good days than bad days but it hasn’t been straightforward.”