The Rotterdam outfit have been tracking the Hungarian international for some time and came in with a contract offer in January.
NAC were powerless to stop him speaking to other clubs due to him having an expiring contract and he agreed terms with Feyenoord this week.
The 29-year-old, who is making steady progress from a shoulder injury, has etched a four-year contract and will move to De Kuip in the summer after seeing out the season with NAC.
Babos is looking to cement his place as Hungary's number one and he will hope the move to Rotterdam will allow him to move ahead of Hertha Berlin's Gabor Kiraly in the pecking order.
Babos' arrival at Feyenoord casts doubt over the long term future of Edwin Zoetebier.
Zoetebier has another season to run on his present deal but has been dislodged as number one by Patrick Lodewijks and could well be offloaded in the summer.
