Jenkins sustained the injury during Cardiff's 35-22 Celtic League defeat at home to Ulster on December 3.
Jenkins, the most-capped Wales player with 129 Test appearances, sustained the injury during Cardiff's 35-22 Celtic League defeat at home to Ulster on December 3.
The 36-year-old is expected to be sidelined for around three months and Wales kick off their Six Nations campaign against Italy in Rome on February 5.
"Gethin is having surgery today on a ruptured bicep, which means he will be out for around three months, but knowing him he will be back before that," said Cardiff head coach Danny Wilson.
"He is one of the most professional men in terms of prehab, rehab and looking after his body.
"Whatever he is asked to do to get that right I know he will do it and I imagine he'll be back before three months.
"It's a big blow for us and it's a big blow for Wales."
If Jenkins takes a full three months to recover from surgery, he is unlikely to play any part in the Six Nations.
Ospreys prop Nicky Smith filled in for him during the recent end-of-year internationals, while the Scarlets' Rob Evans is another option.
Evans started all five games in this year's Six Nations, but missed Wales's recent matches due to injury.