The striker suffered the injury in a cycling accident on holiday and is not expected to recover in time for the trip to St Mary's Stadium on August 12.
Head coach Paul Clement said Llorente's recovery is progressing but the forward could also be unavailable for Swansea's first home game, against Manchester United, the following weekend.
"Fernando's rehabilitating individually - he has not joined the team training yet," Clement said. "But assuming all goes well, we anticipate that happening in the next 10 days.
"I think it is unlikely he will be involved in the first couple of games, but his return will not be long after that. It was always going to be touch and go for the start of the season.
"The most important thing is that we get the arm right. You don't want him to re-fracture it and then end up with Fernando needing surgery, because a six-week injury can then become a six-month injury."
Llorente scored 15 Premier League goals last season to help Swansea stay in the top flight, and the 32-year-old was subsequently linked with a move to Premier League champions Chelsea.