The firm behind the voting system said turnout was at least one million fewer than the government figure.
But the head of the election council hit back, calling it an "irresponsible claim, with unfounded estimations".
President Nicolás Maduro has said that the new assembly will be inaugurated on Friday.
The opposition see the assembly as an attempt by Mr Maduro to cling on to power.
They boycotted the election and also held an unofficial referendum in which they said more than seven million Venezuelans voted against the constituent assembly.
Turnout in Sunday's poll is seen as a key indicator of the level of support the government enjoys.
The government said more than eight million people, or 41.5% of the electorate, had voted.