Kootenai County officials say a construction supervisor found the boys, ages 6 and 3, sitting in the grass near a construction site at 9:10 a.m.
The boys told deputies that their mother ran out of gas and went for help while it was still dark and raining.
Investigators learned that the boys' mother — 27-year-old Shannon Marie Germanton — was given a ride to a gas station between 6:30 a.m. and 7:30 a.m. She told the driver she had argued with her boyfriend and needed to get to a phone to call for a ride.
Officers have been unable to trace her from the gas station.
Sheriff's spokesman Ben Wolfinger said Tuesday afternoon that no car was found near where the boys were waiting, Germanton's cellphone is either off or its reception is obstructed and no one was at home at her last known address.
The children were placed in foster care.
Anyone who knows where Germanton might be is asked to call the Kootenai County sheriff's office at 208-446-1300. She has also used the last name Duval.