According to the Associated Press, the recording hears Mr Glazer say: "We need to descend down to about [18,000 feet]. We have an indication that's not correct in the plane."
The controller told him to fly at 25,000 feet. "We need to get lower," Mr Glazer replied, to which the controller said: "Working on that."
Mr Glazer descended to 25,000 feet, when the controller cleared them for a descent to 20,000 feet - a command recognised by the pilot though he never made it down to that height. The controller asked Mr Glazer to respond but there was no reply.
The couple had taken off at 8.45am local time and controllers made their last contact at 10am.
The plane penetrated Cuba's airspace as it travelled south, with two scrambled US F-16 fighter jets that has followed it trailing off before reaching the island nation.
Cuba reportedly fully co-operated with its diplomatic opponent, sending its own jets to follow the private plane and also allowing a US Coast Guard C-130 aircraft to go through its airspace and did not consider the Glazers' jet to be a violation, according to CNN.
US fighter pilots can be heard discussing Mr Glazer's condition as they followed the Socata jet.
"I can see his chest rising and falling right before I left," one said, in a recording.
"It was the first time we could see that he was actually breathing. It may be a deal where, depending on how fast they meet them, he may regain consciousness once the aircraft starts descending for fuel."
No wreckage has yet been found and search teams are combing the waters for any survivors and it is still unknown if there were any more people on the plane.
Larry and Jane's son Rick Glazer confirmed his parents were licensed pilots.
Jamaica dispatched two aircraft and a dive team, Major Basil Jarrett of the Jamaican Defense Force said, while a US Coast Guard cutter is expected to join today.
Mr Basil added: "An oil slick indicating where the aircraft may have gone down has been spotted in the area where we suspect the crash took place."
Real estate CEO Mr Glazer is described by his company Buckingham Properties as spending "some of his spare time on the ground - gardening around his house with his wife, Jane; and some in the sky - flying his plane. Larry serves as President of the TBM Owners and Pilots Association."
- Independent, Additional reporting by Associated Press