The Brazilian driver is now in intensive care, and unofficial word from the central Budapest hospital AEK is that the operation was to remove a piece of damaged bone from his left eye-socket.
Speculation suggests the surgery took two hours and Massa, 28, is now under observation. Brazilian media reports say the 2008 championship runner-up is expected to make a full recovery.
Reportedly in attendance at the hospital are Massa's brother Eduardo and friend and compatriot Luciano Burti.
McLaren's Heikki Kovalainen, as well as other drivers including Rubens Barrichello and Nico Rosberg, said before leaving the Hungaroring on Saturday that Massa's crash should inspire more talks about how to better protect drivers' heads.
It emerges that in the wake of the death of Henry Surtees, who last Sunday died in a F2 crash as the result of his helmet striking a flying wheel, the drivers discussed the issue of the vulnerability of single seater drivers' heads at length in the pre-Hungarian GP briefing.
"Something needs to be done. Yes, absolutely," said Barrichello.