The newspaper said the wounded officer, eight-year department veteran Aaron Terrazas, 34, was recovering in intensive care at San Antonio Military Medical Center.
"I wouldn't call it a graze," Police Chief William McManus said. "It's a serious wound."
The San Antonio Express-News reported the unidentified 37-year-old suspect, who had "some history" with law enforcement, gave himself up about 5 p.m. CDT. McManus said the suspect would likely be charged with attempted capital murder, the newspaper said.
Police found a handgun in the motel room after the man was taken into custody.
McManus said officers responded to a domestic disturbance call at a Super 8 Motel in downtown San Antonio shortly before 8 a.m. CDT and the suspect shot at them through a door, WFOR-TV, San Antonio, reported. The man then remained barricaded in the motel room until his surrender.
"This came to the conclusion that we wanted it to," McManus said.
The motel's guests were evacuated and students at a nearby Catholic high school were being kept inside. A nearby street was blocked off and Interstate 35 was closed in the vicinity for a time, though the southbound lanes reopened shortly before 1 p.m.