If the Brazilian does not drive, then the likely replacement would be Williams reserve driver Paul di Resta - who is in Budapest this weekend working for Sky TV, stepping in for the unwell Martin Brundle.
Massa went to hospital after feeling dizzy in Friday practice but was released and spent the night in his hotel.
Massa is hoping to be fit enough to race in the Belgian Grand Prix at the end of August, although he does have the benefit of the summer break to recover.
"Following this decision, the team's reserve driver Paul di Resta will drive alongside Lance Stroll for the remainder of the Hungarian Grand Prix weekend". He ended up racing in a vehicle for the Hungarian Grand Prix Qualifying that he had never driven in before.
For the 31-year-old Di Resta, it is an unexpected chance to get back behind the wheel in F1. He was cleared to take part in Saturday's final session before qualifying but ended it early.
"You're very anxious. Nervous but very excited".
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Claire Williams, the team's deputy team principal, added: "It is unfortunate that Paul did not even get this morning's hour which would have been useful, but he is an experienced driver".
"I think if we had to have anyone in our vehicle, you're going to have total confidence with someone like Paul. I haven't driven one of these cars for three-and-a-half years apart from 10 laps I did in a 2014 vehicle and you get thrown into qualifying which is the deepest of all deep, jumping off a cliff and seeing how you can fight for survival". It's a tricky little track and You have to be very committed and confident.
"These cars are top of their game and it's like being back at home driving the best balanced vehicle you've ever driven".
"You imagine, you're so in tune to doing a race weekend, then at the very last minute to not be able to do it, he's genuinely upset".
"I think Felipe Massa is more upset than anything else", said Williams. I'm ready to pounce if he needs a stand in.