Paramedics said that number could rise.
Riot police and heavily armed soldiers quelled the fighting after several hours.
Some family members of the prisoners - forced to wait hours for news of their relatives - tried to force their way into the jail, while others taunted police and threw rocks at the administrative office's windows.
The Reynosa prison, which was designed to house 1,800 inmates, currently holds about 2,200.
Mexico has seen a rise in prison riots this year as Felipe Calderon, Mexico's president, has tried to bring drug-related violence under control.
Despite the deployment of 36,000 troops across the country, violence over smuggling routes is continuing to grow.
More than 3,725 people have been killed in drug-related fighting in Mexico this year.