Images from the scene showed a semi-truck on its side and the bus with front-end damage.
The Gallup Indian Medical center reports it received approximately 37 patients. It was not immediately known what the conditions of the injured were.
Photos on social media showed a truck on its side with the trailer's cargo scattered across the highway, and the Greyhound bus sitting upright with its front end obliterated.
One witness recounted "screams ... coming from the bus", describing it as a "horrible" scene in which survivors attempted to crawl out of the wreckage.
It was due to arrive just after midnight on Friday.
The exact number of fatalities and injuries is not yet clear.
"It was terrible, there was people trying to climb out of the windows of the bus. bystanders trying to help people getting ladders out of their truck to get to windows of Greyhound bus to assist", a witness told KRQE.
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Eric Huff was heading to the Grand Canyon with his girlfriend when they came across the crash.
"We are fully cooperating with local authorities and will also complete an investigation of our own", Greyhound spokeswoman Crystal Booker said in a statement.
The crash happened Thursday afternoon near the town of Thoreau in western New Mexico.
"It was intense", Jones said.
An emergency number has been established by McKinley County Emergency Management for people looking for information about family and friends.
Traffic driving westbound is diverted along a frontage road as police investigate an accident between a semi and a Greyhound bus on Thursday on Interstate 40.