She added, “This shows an unprecedented state of siege underwritten by strategy to destroy electoral institutions and impede the election of a National Constituent Assembly.”

The vote has been widely condemned by Venezuela’s neighbors as a power grab for Mr. Maduro’s leftist movement. The constituent assembly could effectively liquidate any official channels of dissent, leaving opponents with few options beyond street protests.

Just 3.7 million votes had been cast by 5:30 p.m. on Sunday, not long before polling centers closed, Reuters reported on Tuesday, based on official election documents the agency reviewed.

The size of the turnout has been central to the legitimacy of the vote, as both the government and the opposition argue that the other side does not have widespread support. In early July, the opposition held a symbolic referendum against the constituent assembly, and said it drew about 7.1 million voters. That figure was not independently verified.

Smartmatic said in its statement that it had provided voting services for the Venezuelan government since 2004.

“Even in moments of deep political conflict and division we have been satisfied that the voting process and the count has been completely accurate,” the company said. “It is, therefore, with the deepest regret that we have to report that the turnout figures on Sunday, 30 July, for the Constituent Assembly in Venezuela were tampered with.”

Venezuela’s opposition sidestepped the criticism by Smartmatic that it had not participated in the vote or the monitoring of it. Instead, the opposition focused on the government, saying the company’s findings confirmed that the vote had been a fraud.