Under the 1992 constitution, introduced after the dictatorship, a head of state may only serve a single five-year term. But sitting President Horacio Cartes is trying to remove the restriction and run for re-election.
Protesters were photographed smashing in windows of the congress building in the capital, Asuncion and setting fire to the interior. Protesters ransacked the offices of those who had backed the bill, according to reports.
Opposition activist Rodrigo Quintana, 25, was killed by a rubber bullet fired by police when they stormed the offices of the opposition Liberal party. "The police barged in, threw people face down to the ground," according to Liberal Party leader Efrain Alegre, who was also hurt. "They came in aggressively, breaking the doors, it was savagery."
In a statement released on Twitter, President Cartes appealed for calm. "Democracy is not conquered or defended with violence and you can be sure this government will continue to put its best effort into maintaining order in the republic," he said.
Protesters had taken to the streets following a secret meeting of 25 senators - a slight majority of the house - which approved a bill to amend the constitution. The bill must also be approved by the other house of parliament - the chamber of deputies - where President Cartes' party holds an outright majority. A similar proposal had been previously rejected in August. Before the secret vote took place, Congress voted first to change the rules that required lawmakers to wait a year before voting again.
Opponents say the bill will weaken the country's democratic institutions. Opposition senator Desiree Masi said: "A coup has been carried out. We will resist and we invite the people to resist with us."
Paraguay was controlled by military ruler General Alfredo Stroessner, who seized power in a coup, from 1954 until 1989. The new constitution in 1992 created the modern government but there has been a long period of political instability and party infighting, as well as a failed coup attempt.
President Cartes' term is due to end in 2018. The change, if approved, would also allow former President Fernando Lugo to run again. Lugo was ousted in 2012 over his handling of a land eviction in which 17 people were killed.