Tate was a passenger in a golf buggy at a south Wales club on Sunday afternoon when the buggy "lost control", Swansea said.
The 28-year-old former Manchester United trainee was diagnosed with a broken tibia of his left leg after being taken to Morriston Hospital by ambulance.
Tate is one of only three current Swansea players who have played in all four divisions with the club.
He played in the 3-2 win over Hull which saved the Welsh club from relegation from the Football League in 2003.
Tate is in his 10th season with the Swans and has played more than 300 games for the club.
A statement on the club website said: "Swansea City have been dealt a bitter blow with news that defender Alan Tate has broken his leg in a bizarre golfing accident.
"The club's vice-captain was a passenger in a golf buggy that lost control at a local golf club this afternoon.
"Tate was taken to Morriston Hospital by ambulance where he was diagnosed with a fractured tibia of his left leg."
Source: PA
Source: PA