The store, located in the tiny town of Mayhill, about 165 miles (265 kilometers) southeast of Albuquerque, has reportedly been posting such signs for years but drew scrutiny after a traveler stumbled on the latest postings and alerted a television station.
According to a former employee interviewed by local TV broadcaster KOB, the store owner threw out people who were offended by the signs.
"It's been here a long time," Marlon McWilliams told the station. "If you go in there and you offend him, you can't go in there no more.
"He turns lots of people away."
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McWilliams said the owner targeted President Barack Obama and other public figures and sold the signs to customers.
One sign said "Kill Obama" in large letters with the word "care" in small print below it.
The contentious postings have met with a firestorm on social media with many calling for a boycott of the store. Others, however, are defending the store owner's right to free speech.
The owner of the store, which is up for sale, could not be reached for comment nor could the realtor handling the sale.
Staff at a hotel and cafe located next to the store declined comment when reached by AFP.
The Council on American-Islamic Relations (CAIR), an advocacy group, meanwhile issued a statement urging the store owner to remove the signs.
"While everyone has the first amendment right to free speech -- even offensive speech -- we urge the store's owner to remove the sign in the interest of common decency and of our nation's unity at a time of increasing divisions," said CAIR spokesman Ibrahim Hooper.
