Antal Rogan, head of the Fidesz parliamentary group, said the plan needed some "strong adjustments" but would protect children and fight drug trafficking and organized crime.
"We think this is a serious proposal and worthy of support," Rogan said. "Those who oppose drug testing are, knowingly or unknowingly, supporting drugs."
A draft of the bill is expected to be ready in February for debate in parliament, where Fidesz has a two-thirds majority of seats.
Rogan said the test results would be revealed only to parents and a positive test would have no legal consequences for minors.
Fidesz communications director Mate Kocsis, who made the initial proposal last week, had said the tests would also apply to politicians and journalists. But Rogan said legal consultations are needed to see if those groups can be included.
The proposal would also be discussed with experts, teachers and parent associations, Rogan said.
The suggested bill was widely condemned by most opposition parties.
The Socialist Party called the idea a "nightmare," while lawmaker Timea Szabo of the Dialogue For Hungary party suggested instead that parliamentary deputies should take alcohol tests before entering chambers.
Toxicologist Gabor Zacher estimated that there are 800,000 alcoholics and 20,000 drug addicts in Hungary, which has a population of 9.8 million.
"Compared to alcohol, the risk to national health of drug use is insignificant," Zacher told broadcaster InfoRadio, adding that there were 30,000 alcohol-related deaths in Hungary a year and around two dozen drug-related deaths.