The team also promoted receiver Neal Sterling from the practice squad Tuesday.
Grant tore a hip flexor in a loss to Houston on Sunday, and Washington severely strained a hip flexor.
The Jaguars (1-5) also released defensive tackle Ziggy Hood from the injured list after he passed a physical and released quarterback Jeff Tuel from the practice squad. Jacksonville signed rookie receiver Rasheed Bailey and safety Craig Loston to the practice squad.
The 6-foot-4 Sterling, a seventh-round pick by Jacksonville, could see playing time Sunday against Buffalo in London. Receiver Marqise Lee is likely to miss another game with a hamstring injury, and receiver Allen Robinson is dealing with a bruised lower leg.
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