Maduro has previously said that his plan for a national assembly would "modify" the "rotten National Assembly that's now there", and the elected constituents would represent all sectors of Venezuelan society.
Santos says he will continue advocating for a peaceful resolution to Venezuela's almost four months of political upheaval that has left at least 113 people dead.
A candidate in Venezuela's southeastern town of Ciudad Bolivar, 39-year-old lawyer Jose Felix Pineda, was shot dead by assailants who broke into his home overnight, prosecutors said.
Ricardo Campos, who worked as a youth secretary with the opposition Accion Democratica party, was killed during the protest, the head of the national assembly said, according to the BBC.
In the west of the city, national guard troops fanned out, using armored vehicles, rubber bullets and teargas to disperse protesters blocking roads.
Elsewhere, streets in Venezuela have been largely deserted with just a trickle of people heading to vote in the widely criticised elections.
Amid government fears of low turnout, Venezuela's 2.8 million state employees were under huge pressure to vote, including threats of dismissal.
Maduro kicked off voting by casting his ballot in a west Caracas polling station.
Air France and Iberia both announced they were suspending their flights to the country during the weekend vote.
Surveys by Datanalisis, a pro-opposition polling firm, show more than 70 percent of Venezuelans opposed the idea of the new assembly - and 80 percent reject Maduro's leadership.
"In 1999, the first decrees and constitutional acts of the National Constituent Assembly were on regulating the functions of the Congress [Venezuelan bicameral parliament, reformed in 1999], because there were two collegiate bodies at the time, the Constituent Assembly and the Congress, as well as on reviewing the prosecution".
The White House says U.S. Vice President Mike Pence has spoken with Venezuelan opposition leader Leopoldo Lopez about the political unrest wracking the South American nation.
The government said that the new constitution would "guarantee peace and coexistence among all Venezuelans", but the leading opposition coalition, known by its Spanish acronym MUD, has refused to take part in the process.
Neighboring Colombia - a refuge for tens of thousands of Venezuelans fleeing the chaos at home - has said it would not recognize the results of Sunday's election in Venezuela.
The United States has already leveled sanctions at 13 current and former Venezuelan officials, and said further action could be "on the table".
Once one of Latin America's wealthiest nations, Venezuela has spiraled into a devastating crisis during Maduro's four years in power, thanks to plunging oil prices and widespread corruption and mismanagement.
Some in Maduro's administration have broken ranks, most prominently his attorney general.