If two start-ups have their way, videogames might cure more than just boredom. They could also be used to treat attention-deficit hyperactivity disorder.

Akili Interactive Labs Inc. of Boston, formed by start-up-creating firm PureTech Ventures, and San Francisco company Brain Plasticity Inc. are seeking Food and Drug Administration approval for a videogame treatment they hope clinicians will turn to before prescribing medicines for ADHD.

The...