NFL legend John Elway was right in the middle of the golf cart accident that left his longtime friend on life support, and he might have been driving ... TMZ Sports has learned.

Law enforcement and other sources tell us Elway was on the golf cart Saturday evening in La Quinta, California when Jeff Sperbeck -- John's business partner and former agent -- fell off the back of the moving vehicle, rolled and hit his head on asphalt ... suffering a traumatic injury.

John, Jeff and their wives had been at the Stagecoach music festival earlier in the day in nearby Indio, and were leaving a post-party when the accident happened. We're told John is the one who made the call to 911 immediately after the accident ... and when paramedics arrived, Jeff was unconscious but still breathing. The 61-year-old was transported to Desert Regional Medical Center in Palm Springs.

Now, multiple sources tell us John was actually driving the golf cart, however, we have not been able to confirm that with Elway. There's no indication he did anything wrong.

No one else was injured in the accident -- which happened on the grounds of a private golf community -- and we're told cops did not respond to the scene.

(http://www.autoadmit.com/thread.php?thread_id=5718657&forum_id=2#48893070)