Poulter, who finished joint third at Muirfield last year, underwent an MRI scan on Monday but declared himself fit enough to compete after receiving the results.
The 38-year-old suffered the injury last week during the Scottish Open at Royal Aberdeen, where he missed the cut by three shots after running up a triple-bogey on the final hole of his second round.
An MRI showed no tendon damage. "I've had a ganglion cyst on the wrist in the past," said Poulter. "By jarring it, some fluid has come out of the joint and into the cyst sac there. It's touching on the nerve so I'm just getting a little bit of pain. I'll be chewing some anti-inflammatories and some painkillers for the week but it's good news, there's no tendon, joint or bone damage."
Mark Calcavecchia, the 54-year-old American who won the Claret Jug in 1989 at Troon, has withdrawn from the Open. He has been replaced by Canada's David Hearn, who will make his major debut. The First reserve is now America's Daniel Summerhays.