A Mexican drug cartel accused of kidnapping four Americans and murdering two of them has said the killers acted alone and will be handed over to authorities, local media reported on Thursday.

Several Mexican news outlets published a letter allegedly issued by the Gulf Cartel, in which the organization said it "strongly condemns" and "apologizes" for the kidnappings and murders, which were carried out without the authorization of the group's leaders.

The four Americans had driven into northern Mexico in a white minivan last week before they were shot at and abducted by gunmen in the border town of Matamoros.

Two were found dead on Tuesday, a third was found with gunshot wounds and another was unharmed. A Mexican bystander was also killed in the gunfire.

Mexican authorities had suggested that the abduction was the result of a "mix-up" rather than a targeted attack.

The victims visited Mexico because one of them was planning to have cosmetic surgery, a Mexican official said.

Alongside Thursday's letter, local media also published a photograph that shows the handwritten note placed next to five men lying face down with their hands bound.

Mexican authorities have not confirmed whether these men were the ones suspected of the murders and kidnappings, or if they have been handed over to authorities.

The prosecutor's office reported the arrest of one suspect, who it said had been guarding the kidnapped Americans.

The two survivors were returned to the United States via a land border crossing between Matamoros and Brownsville, Texas.

The bodies of the two dead Americans were repatriated on Thursday, authorities said.