The Scarlets hooker may need surgery, but national coach Warren Gatland hopes that he will be fit for the start of the tournament.
"Matthew has had a bit of a neck problem for a few weeks," he said.
"He is getting injections today, and we're hopeful he will be alright in the next week or two.
"An operation is a definite possibility but we are hoping it definitely won't be before the World Cup. If he doesn't improve, he may need something after the World Cup.
"There is a bit of nerve pressure, and we're hopeful the injections will help. It is mainly affecting him when he is trying to sleep."
Cardiff Blues flanker Sam Warburton is likely to captain Wales in Rees’s absence.