That would mean a second missed opportunity for officers to stop Salvador Ramos before the May 24 rampage inside the school that killed 19 children and two teachers.
Officials said school district police also drove past Ramos without seeing him in the building's car park.
The unidentified officers, one of whom was armed with an AR-15-style rifle, said they feared hitting children playing in the line of fire outside the school, Chief Deputy Ricardo Rios of nearby Zavalla County told the newspaper.
The chance of stopping Ramos passed quickly, perhaps in seconds, Rios said.
Police at the scene did not move to neutralise the gunman for more than an hour, even as anguished parents outside the school urged officers to go in.
Messages from The Associated Press to Rios and the Zavala County Sheriff's Office have not been returned. The Zavala County sheriff's officials responded to the shooting in support of Uvalde and Uvalde County officers.
Uvalde police officials agreed on Friday to speak to the committee investigating the attack, according to a Republican politician leading the probe who had begun to publicly question why the officers were not co-operating sooner.
"Took a little bit longer than we initially had expected," Dustin Burrows said.
On Thursday, Burrows had signalled his impatience with Uvalde police, tweeting that most people had fully co-operated with their investigation "to help determine the facts" and that he did not understand why the city's police force "would not want the same".
He did not say which members of the department would meet with the committee, which is set to continue questioning witnesses in Uvalde on Monday.
Uvalde police did not reply to messages seeking comment.
Weeks after one of the deadliest school shootings in US history, law enforcement officials have stopped providing updates about what they have learned about the shooting and the police response.
Their silence comes after authorities gave conflicting and incorrect accounts in the days after the shooting, sometimes withdrawing statements hours after making them.
Officials have also not released records sought under public information laws to media outlets, often citing broad exemptions and the ongoing investigation.
The state House committee has interviewed more than a dozen witnesses behind closed doors so far, including state police, school staff and school district police.
The list of witnesses provided by the committee has not included Pete Arrendondo, the Uvalde school district police chief, who has faced criticism over his actions during the attack.
Burrows defended the committee interviewing witnesses in private and not revealing their findings so far, saying its members want an accurate account before issuing a report.
"One person's truth may be different than another person's truth," the politician said on Friday.
Since the shooting, Republican leaders in Texas have called for more mental health funding but not new gun restrictions.
