The Spaniard was involved in a cycling accident while enjoy a summer vacation in Italy.
Llorente is now in a race to be fit for the start of the Premier League season in six weeks’ time.
The former Sevilla and Juventus forward was riding a bike as he picked up the injury, but it is unconfirmed whether he was involved in a collision or simply fell off.
A club statement confirmed: “Swansea City can confirm striker Fernando Llorente has fractured a bone in his lower arm.
“The 32-year-old suffered the accident while cycling on a family holiday, and despite it not being a serious break that requires an operation, it does mean the club will have to tailor his pre-season training accordingly.
“The club remain optimistic that he will be available for selection for the first game of the Premier League season against Southampton on Saturday, August 12.”
Paul Clement’s side are due to return to pre-season training imminently.
Llorente helped keep Swansea in the Premier League last season after plundering 15 goals, turning their fortunes around.
The veteran is expected to lead the line for the Swans once again next season, despite the arrival of Tammy Abraham on loan from Chelsea.
Llorente was linked with a move to Stamford Bridge in January and the Swans were braced for further approaches this summer.
But the latest news may all but end any interest in Llorente, as he will miss the majority of pre-season.