"Gethin is struggling and could be out for four to five weeks," said Gatland at the tournament launch in London.
"Gethin is a huge loss for us for potentially the first two games of the tournament. He's been playing exceptionally well for the Blues in the last month.
"He's a world-class player. He's got a very good rugby brain and is able to assess things on the field. He's almost like a coach on the field. You get a lot of information from him at half-time, what's going right or wrong, and tactically where we need to change."
There is more bad news for Wales on the injury front, with fly-half Rhys Priestland also expected to miss the tournament opener against Ireland in Dublin.
The Scarlets star was a revelation at the recent World Cup, seeing off both Stephen Jones and James Hook for the No10 shirt in New Zealand.
Like Jenkins, Priestland has a knee injury, while first-choice blindside Dan Lydiate is struggling with an ankle problem.
Wales are already without inform second row Luke Charteris for the entire campaign and Alun Wyn Jones for the start of the 6 Nations but Lions centre Jamie Roberts and full back Lee Byrne could be fit enough to feature in Dublin following knee and thumb problems respectively.
"Rhys will be struggling for the first game. We're assessing him day by day," added Gatland.
"Rhys was probably the find, if not the best, fly-half at the World Cup. He really does fit the type of game we want to play.
"His kicking game and vision are excellent. He stretches opposition defences as well because he's not afraid to run.
"There's a very good balance to his game. You can't underestimate the space he created for our midfield at the World Cup.
"He was one of the reasons we saw some really big performances from Jamie Roberts, Jon Davies and Scott Williams. He will be a big loss for us.
"But there's experience to come in with James Hook and we may look at Gavin Henson in that position too.
"Dan Lydiate is touch and go but we're hoping Jamie Roberts will be OK."
Wales face Ireland on Sunday, February 5 and host Scotland at the Millennium Stadium a week later. They then travel to Twickenham to take on England on February 25 before returning to Cardiff for the games against Italy and France.