The government of Tamaulipas state, where Reynosa is located across the border from McAllen, Texas, said another three people died late on Thursday in other confrontations.
The marines were responding to reports of armed men when they came under fire early yesterday.
Two suspects were killed in the first confrontation, and two escaped. Three other suspected drug gang gunmen opened fire on marines and were killed in a second shootout yesterday.
The driver of their taxi was also killed.
The dead gunmen were believed to be part of a Gulf Cartel wing commanded by Julian Loisa Salinas, known as Comandante Toro. Reynosa has long been dominated by the Gulf drug cartel.