There were reports that three protesters and a child were also killed in Kailali district on Monday.
Government Administrator Raj Kumar Shrestha said thousands of protesters attacked with spears, knives and stones police officers trying to enforce a curfew.
The wounded were taken to the regional hospital and the death toll was expected to rise, Al Jazeera reported.
A new national constitution presented in parliament on Sunday aims to restructure Nepal as a federal state made up of seven provinces.
The protesters argue that the new borders will discriminate against historically marginalized communities.
Tharu minority
Monday's violence erupted as members of the Tharu ethnic minority had demonstrated to demand their own separate province.
Home Minister Bam Dev Gautam said the government would send troops to the area to try to bring the situation under control, and appealed for calm.
"At a stage when we are heading to a new era ... we must call on people to not be involved in activities that disturb social harmony and ignite ethnic hatred," Gautam told politicians.
He said the area had been declared a "riot-stricken zone" and the army deployed.
Nepal's main parties struck a breakthrough deal on the charter in June after April's devastating earthquake helped end years of wrangling that had left the impoverished Himalayan nation in political turmoil.
Work on a new national constitution began in 2008, two years after the end of a decade-long Maoist insurgency that left an estimated 16,000 people dead and brought down the 240-year-old Hindu monarchy.