Cardiff Blues flanker Sam Warburton, who skippered Wales in their defeat to the Barbarians two months ago, is lined up to keep the job at Twickenham next Saturday.
Scarlets hooker Rees led Wales throughout last season's international schedule, including the RBS 6 Nations Championship.
But he has been suffering with a neck injury in recent weeks, and Wales coach Warren Gatland admits surgery is a possibility.
Gatland, though, hopes any operation that might be required would not take place until after Wales' World Cup campaign, which begins against South Africa in Wellington on September 11.
"Matthew has had a bit of a neck problem for a few weeks," said Gatland.
"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.
"Ideally, you would like him to play in the warm-ups because you don't want him to go in cold.
"He is getting frustrated because he hasn't been able to do much in the last couple of weeks."
Wales will play three World Cup preparation fixtures this month, with a return match against England in Cardiff on August 13 followed by Argentina visiting the Millennium Stadium seven days later.
Gatland intends announcing his final 30-man World Cup squad for New Zealand on August 22.
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