According to Galveston Beach Patrol Dispatch, a child's body that meets the description of one of the swimmers was found at about 2 a.m. by a group of four people walking on the beach.
They pulled the boy to the shore then called 911.
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"Caller just pulled an unconscious person, young teenager, out of the water. Blue shirt, red shorts. Not moving. EMS is going to be in route," a dispatcher was heard reporting on radio traffic. "Male meets the description of one of the missing children from two days ago," another official said.
After search teams determined the boy's clothing and physical description matched the twins, the boys' family was notified.
Lt. Austin Kirwin with Galveston Beach Patrol said the boys are from Houston. They reportedly did not know how to swim.
Search crews plan to return to the same area on Tuesday to search for the second teen.
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The boys disappeared at about 4:30 p.m. Sunday when they were last seen swimming on the west side of Pleasure Pier, about 20 yards from the shore.
Their family called 911, saying they last saw their children in water about waist-to-chest deep. The boys are around 4-feet tall.
Along with beach patrol, police, fire and EMS all responded to search the area. Search crews were launched and the U.S. Coast Guard flew its helicopter overnight into Monday in hopes of finding the boys.
The search for the boys continued Monday morning along the shore line and in the water. KTRK crews spotted officials on jet skis searching and diving.
Family members of the missing teens were out at the search site but too distraught to speak on camera. They did, however, share a photo of the boys.
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Kirwin warned about strong currents and waves this time of year in Galveston, advising that people watch their kids closely and look out for jetties and piers in the water.
A lateral current tends to form in the Gulf waters in Galveston.
"Whenever that current hits an object out in the water, such as these jetties, it creates a rip current that forms, which pulls you out to sea away from shore," Lt. Kirwin said. "So, a lot of times, things will get caught in that rip current and they could get stuck on the rocks. They could just get kicked farther out to sea."
On Monday, several people had to be rescued from rip currents in Galveston.
KTRK was told beach patrol saved three girls and a boy, and that someone who tried to help during that rescue needed saving, too.
WATCH: Galveston Beach Patrol's 4:30 p.m. update a day after twins disappear
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Galveston Beach Patrol's 4:30 p.m. update a day after twins disappear
Beach patrol said that it's running mobile patrol vehicles and providing 24/7 on-call service.
It added that stationed lifeguards will start working the beach on Saturday, March 11. Lifeguard tryouts will also be held that day.
If you panic and try to fight the current, chances increase that you can drown.
Remember to swim within your limits and near a lifeguard if possible.
Non-swimmers and children should always wear life jackets when in or around the water.