A Mexican cartel has dumped five of its own members in the street for police to take into custody along with a letter apologising for accidentally kidnapping American tourists.

In a hand-written note apparently left alongside the five men found with their hands tied, the Scorpions Group faction of the notorious Gulf Cartel apologised to the four Americans and their families, the Mexican woman who died in the cartel shootout, and the town of Matamoros where the kidnapping took place.

“We have decided to turn over those who were directly involved and responsible in the events, who at all times acted under their own decision-making and lack of discipline,” the letter read.

The cartel members responsible for the botched abduction, it said, went against the cartel’s rules, which it claimed include “respecting the life and well-being of the innocent”.

A Mexican state security official said that five men had been found tied up inside one of the vehicles that authorities had been searching for, along with the letter.

Last Friday, the four Americans crossed into Matamoros from Texas, reportedly so that one of them could have cosmetic surgery.

At around midday, the white minivan they were travelling was attacked as they drove through Matamoros.

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

The two Americans who died were identified as Zindell Brown and Shaeed Woodard, while the two survivors were named as Eric Williams and Latavia McGee.

A Mexican woman, Areli Pablo Servando, 33, was also killed, apparently by a stray bullet.

Mexican officials handed the bodies of the two dead men to US officials in Matamoros on Thursday afternoon, Reuters reported.

Mexican state prosecutors on Thursday said they had seized an ambulance and cordoned off a medical clinic in Matamoros where the victims are believed to have been treated after the shooting.

A cousin of Mr Williams said his family feels “great” knowing that he was alive but does not accept any apologies from the cartel blamed for kidnapping the Americans.

“It ain’t gonna change nothing about the suffering that we went through,” Jerry Wallace told the Associated Press on Thursday. Mr Wallace, 62, called for the American and Mexican governments to do more to address cartel violence.

Drug cartels have been known to send threatening messages to intimidate rivals and the authorities, but they have also occasionally sought to smooth over situations that could affect their business.

With Mexican National Guard troops and Army special forces patrolling Matamoros after the deaths of the US citizens, Mexican security analyst David Saucedo said: “It is very difficult right now for them to continue working in terms of street-level drug sales and transferring drugs to the United States.”

The cartel, he said, “are the first ones interested in closing this chapter as soon as possible”.

Handing over alleged cartel suspects to police is also not without precedent. Mr Saucedo suggested that a cartel leader might have authorised the attack then regretted it and decided to offer sacrificial lambs to police.

In 2008, drug traffickers in Michoacan lobbed hand grenades into a crowd celebrating Mexico’s independence, killing eight.

Days later, authorities arrested three suspects, but it later emerged that they had been kidnapped by a cartel, beaten into confessions implicating a rival group and turned over to police.