The Scarlets hooker has already been ruled out of Wales' opening tournament warm-up game against England next Saturday, with flanker Sam Warburton set to take charge.
Wales coach Warren Gatland admits surgery is a possibility for Rees, who has won 46 caps and was appointed Wales skipper last November.
But the 30-year-old is due to undergo injections in the hope it will relieve pressure on his neck, with any operation not taking place until after the World Cup.
His condition, though, is a worry for Gatland, especially as Ospreys hooker Richard Hibbard, who provided bench cover for Rees throughout last season, is still recovering from shoulder surgery and might not be available for any of Wales' three warm-up fixtures.
It leaves the experienced Huw Bennett in pole position to start at Twickenham, although Gatland's squad also includes Newport Gwent Dragons hooker Lloyd Burns and Rees' Scarlets understudy Ken Owens.
"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 all right in the next week or so.
"It has been a bit of an irritation for him because he hasn't been able to do much contact.
"An operation is a 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 are hopeful the injections will help.
"We've rested him in the last couple of weeks, and we are hoping that by having the injection there will be a significant improvement.
"It's not actually while he is training or playing sometimes - it's when he is resting and trying to get sleep at night. We are trying to relieve that pressure.
"Ideally, you would like him to play in the warm-ups because you don't want him to go into the World Cup cold.
"He is getting a little bit agitated at the moment because he hasn't been able to do much in the last couple of weeks and he can see the players going away from him a little bit.
"He is hopeful the injections will work and he will be back into training pretty quickly."
The Wales players reported for duty today following some time off after the second of two rigorous training camps in Spala, Poland.
And after suffering landslide World Cup warm-up defeats against England in 2007 (62-5 at Twickenham) and 2003 (43-9 in Cardiff), they will not lack an incentive in home and away clashes during the next 12 days.
Gatland added: "There will be no lack of motivation for the players over the next two weeks.
"It is England versus Wales, but more so, I have seen how motivated these players are to want to make an impression and get selected for the World Cup squad.
"We are conscious of what happened in 2007 and in 2003, when England gained comprehensive victories.
"It is important we pick two strong sides for these games and give a good account of ourselves.
"I can't speak highly enough of the players and the way they approached the two training camps in Poland. We pushed them very hard, and the players really responded.
"We know how physical and tough our (World Cup) pool is going to be, but we are also excited that if you come out of that pool you feel you are in good shape for whatever team you might face in the quarter-final."
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 to announce his final 30-man World Cup squad for New Zealand on August 22.
Wales' opening game is against reigning world champions South Africa in Wellington on September 11, followed by an appointment with a dangerous Samoa side seven days later. Their pool also includes Namibia and Fiji.
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