The transaction is expected to close at the end of April.
MOL, which is among the biggest backers of Hungarian football, said it could "do more for the success of Újpest FC as an owner".
"A strong Újpest can only strengthen Hungarian football further," said Péter Ratatics, MOL's executive in charge of retail services, who will represent the company at the club.
Újpest FC has been owned by the Belgian businessman Roderick Duchatelet for more than a decade.
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