The Washington Redskins quarterback suffered a horror double break leg during a match against the Houston Texans last month.
And reports now suggest the 34-year-old's career could be in doubt following an infection related to the surgery.
It is understood that Smith has gone under the knife several times in a desperate bid to combat the problem.
But the player's club posted a worrying statement to their Twitter page on Thursday.
It read: "On behalf of Alex Smith, we appreciate all of the concerns and prayers over the injury he incurred on November 18th against the Houston Texans.
"Although this is a serious injury, Alex and his family remain strong.
"We would ask that everyone please honour the Smith family's request for privacy at this time."
Smith suffered a broken tibia and fibula during the Redskins' 23-21 defeat.
Coach Jay Gruden ruled him out for the rest of the season after he collapsed to the ground when his ankle got stuck in the turf.
Before he was aware of the infection, Gruden said: "He's got a long way to come back but physically and mentally he'll be fine, but it's hard to watch that happen to any athlete."
As he left the pitch in agony on a stretcher, players from both teams dropped to their knees in respect for the 34-year-old.
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