Hungarian television channel MTV reported that Ujhelyi Istvan, a socialist MEP currently campaigning for re-election, had promised to break down the anti-migrant fence along the country's southern border.
"Ujhelyi István promised in public, that if MSZP, the Hungarian socialist party wins, he would break down the fence immediately, the report said.
"The socialist MEP spoke at a campaign event at Sopron, saying they (his party) are the only guarantee of making a Europe without fences."
Ujhelyi, however, denies making the comments. His spokesperson told Euronews that he did not make the statement.
The MEP released a Facebook video saying that "the flunkey media of Fidesz has lied again".
He said his real comments were: "Dear Sopron people, on the May 26, go to vote because if you stay at home, you'll support Fidesz and the European far-right.
"If you would like a unified Europe without fences, you should vote for the list of the Hungarian Socialist party, the party's 'dialogue' and a socialist Europe."
Hungary's right-wing government built a 175-km-long barrier on its border with Serbia at the height of the migrant crisis in 2015.
It later constructed barriers on sections on its border with Croatian that are not separated by the Drava river.