Naim Rasool Mohammed, 32, was arrested in a creek bed hours after police issued an alert saying he was suspected of abducting the two boys, ages 3 and 5.
According to a police report, Mohammed approached the boys and their 22-year-old mother as they were walking to school and forced them into his vehicle by threatening her with a brick. The woman eventually managed to jump from the vehicle and flag down a Dallas County constable, who notified police but did not give chase.
Mohammed also tried but failed Monday to take his youngest child, a 1-year-old, from another location, Police Deputy Chief Craig Miller said at a news conference.
The boys' mother told officers that Mohammed had threatened to kill the children, and that he had previously threatened to them and her, police said. Miller said there was a "continuing family violence situation" between Mohammed and the boys' mother.
Hours after the boys were taken, authorities received a call from a woman who said her son had drowned her grandchildren and that she had the children, who were unresponsive, in her car awaiting paramedics, Dallas Fire-Rescue spokesman Jason Evans said in a statement, according to the Dallas Morning News. He did not identify the caller.
The boys were pronounced dead at a hospital.
Mohammed was being held at the Dallas County Jail. Online records, in which his last name is spelled "Muhammad," showed he did not have an attorney representing him.
Miller said investigators were trying to determine where the two boys were drowned.
Mohammed was previously convicted of violent behavior.
In September 2009, he pleaded guilty to aggravated assault with a deadly weapon, a second degree felony, and received five years' probation. According to court records, which also spell his last name Muhammad, the charge stemmed from a complaint from his sister that he had struck her in the head with a hammer.
Mohammed also has convictions for burglary of a vehicle, theft by check and marijuana possession, according to the Texas Department of Public Safety's online data base.