Dee, 30, missed the end of last season and the summer Tests against South Africa and Australia but was expected to return at the start of this campaign.
Dragons head coach Dai Flanagan says Dee, who has played 51 Wales internationals, suffered a “little setback” and is not expected back until December.
Young Cardiff hookers Evan Lloyd and Efan Daniel, who both toured Australia, are struggling with injury, their captain Liam Belcher is definitely missing, and Ospreys number two Sam Parry was out of favour in the summer.
With Dewi Lake and Ryan Elias the experienced hookers available to Gatland, young Dragons front-rower Brodie Coghlan, 23, could be an another Wales option after impressing for the Dragons at the start of the season.
“We would be ecstatic if any of our players gets picked but probably more so with Brodie given the work and effort he has put in,” said Flanagan.
“Not long ago he was up in [North Wales regional team] RGC on loan fighting to be a professional rugby player, but it’s one step at a time for him.
“We didn’t necessarily think we would see this level of impact but what we did see was somebody who took his job very seriously over the summer.
“He changed his body shape, reflected on what he needed to do physically, and you can see a very fit player.”
Dragons have a loose-head prop injury crisis before the United Rugby Championship (URC) visit of Benetton to Rodney Parade on Saturday evening.
Rhodri Jones has been ruled out for the rest of the year because of a knee injury leaving Rodrigo Martinez as the only Dragons number one who is fit, with Ospreys on-loan prop Cam Jones acting as back-up.
With Josh Reynolds also sidelined, Dragons have moved to sign an overseas prop until January and he is set to arrive this weekend.
“We have a loose-head coming in very soon to help us out,” said Flanagan.
“We have an injury crisis and the Ospreys were very good in allowing Cameron to come to us for this block.
“Josh is close to a return but isn’t contact-ready yet, while Rhodri will be out for a long time with an injury he picked up late in the Sharks game.
“Dylan Kelleher-Griffiths has had ankle surgery and Jordan Morris had knee surgery, so that leaves us with Rodrigo as the only fit loose-head on contract.”