One described how a plane rented from Damojh airlines had completely dropped off radar systems around eight years ago, while another alleged poor maintenance.
Mexican authorities say they will carry out a safety audit of the company, BBC News reports.
Three women survived the crash near Havana airport - Cuba’s deadliest air disaster in more than 30 years.
Cuban authorities have launched an investigation into the accident, as rescue workers continue to comb through the wreckage site around 20km (12 miles) south of Havana.
One of two the aircraft’s black boxes has been recovered and is said to be in good condition.
Allegations of previous safety complaints have emerged against the Mexican-based Damojh company, which leased both the Boeing 737 and its crew to Cuba’s flagship state Cubana airline.
The head of Guyana’s civil aviation body, Cpt Egbert Field, told the Associated Press news agency that the same aircraft - which was nearly 40 years old - had been barred from using Guyanese airspace last year after authorities found its crew were overloading luggage on flights in Cuba.
In one instance, the news agency reports, Guyanese authorities had discovered suitcases stored in the plane’s toilets.
Meanwhile, a retired pilot for Cubana wrote on Facebook that another plane rented by his airline from the same company had briefly dropped off radar for unspecified reasons while over the central Cuban city of Santa Clara in 2010 or 2011.
The captain and co-pilot of the flight were later suspended for "problems and serious lack of technical knowledge," said Ovidio Martinez Lopez, who worked for Cubana for more than 40 years.
Lopez said "many flight attendants, flight attendants and flight safety personnel" had refused to fly on certain Cubana planes over the years.
Another pilot who used to work for Damojh told Mexican newspaper Milenio he had complained about a lack of adequate maintenance of planes.
"I experienced several incidents at this company, like engine failure or the electrical system went when we took off from Mexico on one occasion", Marco Aurelio Hernandez was quoted as saying.
Damojh has yet to comment on the allegations.
