Eddie Guerra, Hidalgo County sheriff, told the McAllen Monitor newspaper that the dead appeared to be two infants, a toddler and a woman in her early 20s.
Guerra said on Twitter that the FBI would investigate the deaths.
The sheriff said that they may have been dead for a few days by the time they were discovered on Sunday in Las Palomas Wildlife Management area near the Rio Grande River.
The newspaper reports that area is known locally as El Rincon del Diablo, or the Devil's Corner, and is often used for irregular border crossings.
Constable Larry Gallardo told the newspaper that the terrain can be difficult to traverse.
"There's a lot of brush," he said. "It's like ranchland, there's no difference."
The names of the four have not been released and authorities are working to determine their country of origin.
Irregular crossings
The Rio Grande Valley sector of the Border Patrol, which includes the area where the four bodies were found, has experienced an unprecedented number of apprehensions involving people entering the country between official ports of entry.
Over a seven-month period ending in April, Border Patrol agents had apprehended more than 164,000 people along the border. The number surpasses the total number of apprehensions in all of fiscal year 2018.
Raices, the largest immigration legal services non-profit organisation in Texas, blamed the anti-migrant policies of US President Donald Trump for the latest deaths.
"Trump's tried so hard to stop people from getting to the border. This is the result," Raices said on Twitter.
"How many more will die before we stop criminalising newcomers and instead treat them with humanity?" it said, adding that no one fleeing for their lives should have to put their lives in additional danger.
Trump declared a national emergency at the US-Mexico border in February, describing the irregular migration as a "growing security crisis".