The grisly discovery - including one body that had been stuffed inside a kitchen garbage bag - was made by anthropologists at a cemetery in Falfurrias, Texas, 80 miles (130 km) from the state's southern border with Mexico.
"These people deserve a proper investigation and a proper burial," Senator Juan Hinojosa, a Democrat, told Reuters.
Hinojosa said he and the local district attorney would meet next week to request an inquiry by the Texas Rangers, the Texas Department of Public Safety's investigative arm.
The findings were first reported by the Corpus Christi Caller-Times on Thursday.
Local authorities are launching their own investigation. They believe the bodies were buried by the Funeraria del Angel Howard-Williams funeral home and are not anticipating criminal charges, said Brooks County Chief Deputy Sheriff Benny Martinez.
The county has paid the funeral home for at least 16 years to deal with the remains of immigrants who die as they cross on foot in the desert-like conditions, according to the Caller-Times.
A funeral home official told the Caller-Times the company would not confirm it was responsible for the burials. Calls by Reuters to the home were not returned on Saturday.
About 120 bodies of immigrants are found each year in Brooks County, Martinez said. Some 235,000 illegal immigrants were apprehended last year near the Mexican border with the United States, according to U.S. Immigration and Customs Enforcement.
The Brooks County graves were found earlier this month by anthropologists working to identify immigrants who die on their way through Texas from Mexico. Officials have not determined how many bodies were in the graves.
The graves held body bags containing more than one body, corpses buried with no bags at all, and other remains that were missing limbs, the Caller-Times said.
The bodies are estimated to have been buried from 2005 to 2009, said Dr. Lori Baker, a forensic anthropologist at Baylor University and one of the researchers who discovered the graves. Her group, the International Consortium for Forensic Identification, is working to identify the bodies.
"These are all people," she said. "Their families deserve to know what happened to them. And they deserve human dignity."