The crash shut down the westbound lanes of Interstate 40 near Thoreau near mile marker 50, police said.
New Mexico State Police did not say how many people died or how many were injured in Thursday's crash.
Debris from the wreckage littered the stretch of Interstate 40, with the semi-trailer spilt open, the truck's cab upside down, and in the middle, the front end of the shredded shell of the Greyhound bus completely ripped off, reports CBS News correspondent Don Dahler.
It was traveling between Albuquerque and Phoenix when state police said it crashed along along Interstate I-40 in McKinley County.
He said 20 people were brought in, four of whom were children.
At least seven people were killed, and numerous 49 passengers aboard the Greyhound bus were injured, although authorities couldn't immediately provide an exact count of how many were hurt or their conditions.
The National Transportation Safety Board said a team of its investigators would arrive on the scene early on Friday.
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Passengers were trapped on board the bus, screaming as rescuers tried to get them out through windows, according to one witness who passed by the scene.
As of Thursday night, eight people were still at the medical center and seven had been discharged. "Our thoughts and prayers are with everyone as we continue to give our support to all affected". One was in critical condition and two were in serious condition, Gaines said.
A large number of emergency services staff can be seen in attendance.
Hospitals in Gallup said they have been told to be ready to receive eight patients. The rest are in stable condition. Part of the side of the bus was torn off, he said.
The crash happened Thursday afternoon near the town of Thoreau in western New Mexico.
"It just stays in your mind", he said.