The Newport Gwent Dragons star - part of an outstanding Wales back-row alongside skipper Sam Warburton and number eight Toby Faletau - suffered ankle ligament damage during last weekend's crucial Pool D victory over Samoa.
He looks unlikely to play again in the group phase - Wales have two pool fixtures left - with a potential quarter-final against Ireland on October 8 thought to be his comeback target.
"The medical team are working around the clock, and Dan is getting up every two hours, icing his foot," Wales assistant coach Robin McBryde said.
"Dan is professional to the end. He's a big part of this team. With him, Sam and Toby in the back-row, as we saw against South Africa, they are quite a unit together."
Wales are set to reassess their injury situation after next Monday's clash against Namibia in New Plymouth.
Full-back James Hook's shoulder injury will be reviewed later this week, while it is hoped that back-row forward Ryan Jones, who has yet to play in the tournament because of a calf muscle problem, will participate in full training tomorrow.
If Wales decide to summon a back-row replacement, then it looks likely to be a choice between Ospreys pair Jonathan Thomas and Justin Tipuric, plus the Scarlets' Rob McCusker.
Three other possibilities are all injured, with Melbourne-based Gareth Delve having undergone knee surgery, flanker Martyn Williams recovering from a broken arm and Josh Turnbull sidelined by shoulder trouble.
"We are just going to use as much time as we've got to find out a little bit more about Dan and James to give them as much time as possible," McBryde added.
"We haven't got to rush into any sort of decision. We have got players who can cover the back-row.
"(Lock) Alun-Wyn Jones could move into the back-row, he is a realistic option for us. We are not down to the bare bones yet.
"We hope Ryan will be able to take a full part in training tomorrow. Slowly but surely, we are getting back there. We are hopeful these guys will pull through.
"I think we have got to be realistic. If we pick up any more injuries following the game on Monday, we will have to reassess the situation and move pretty quickly then.
"We will see how we get through the game.
"Gareth (Delve) has had an operation on his knee, which has ruled him out.
"We have discussed options. We've looked at all the regional games in Wales at the weekend, and the Ospreys, in particular, with Jonathan Thomas and Justin Tipuric there, and you've got Rob McCusker at the Scarlets.
"We are keeping an eye on how things are going at home. Josh Turnbull is injured and ruled out for now, and Martyn (Williams) with his arm.
"We are taking stock, and that is another reason why we are not rushing into premature decisions."
Fly-half Stephen Jones, meanwhile, looks set to start against Namibia after being laid low by a calf injury since early August.
Should Jones feature in New Plymouth, he will win his 101st cap and break the Wales appearance record he currently jointly holds with Gareth Thomas.
And McBryde ruled out any hint of complacency against a Namibian side that has conceded 98 points in two games so far.
He added: "I don't think there are any easy games in this competition. The so-called lesser nations have all put up a good fight, and I don't expect anything different on Monday.
"Namibia have got a good strong scrum. They run hard, and it is a typical South African type of mentality. As ever, the contact area is going to be a huge issue.
"(Jacques) Burger is the main man, along with their number eight. They are combative at the breakdown, which is a key area. If you don't get quick ball from there, your attack is very much stifled.
"We like to play off the front foot, with momentum and continuity, and hopefully we can do that on Monday."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.