By Humeyra Pamuk and Sarah Morland

KINGSTON (Reuters) - U.S. Secretary of State Marco Rubio on Wednesday said that he would engage with Jamaica over its employment of Cuban doctors to better understand the program there, after Washington threatened to halt visas for officials tied to these programs.

Rubio has accused officials involved of human trafficking and labor exploitation, accusations denied by Cuba and many Caribbean leaders, who say the Cuban medical programs are essential and comply with international labor laws.

"Jamaica has a deficit in health personnel," Jamaican Prime Minister Andrew Holness told a joint press conference with Rubio. "We ensure that they are treated within our labor laws and benefit like any other worker."

"Any characterization of the program by others certainly would not be applicable to Jamaica," he added.

Earlier this month, Jamaican Foreign Minister Kamina Johnson Smith told reporters her government was concerned about the accusation and the presence of some 400 Cuban doctors, nurses and medical technicians was important for Jamaica's healthcare system.

"Their presence here is of importance to our healthcare system," she said, pointing to 400 doctors, nurses and medical technicians currently working in the country.

Elsewhere in the region, other Caribbean leaders have rejected the accusations of labor exploitation and some have said they would rather give up visas than the medics.

As well as targeting countries receiving Cuban medics, who have since the country's 1959 revolution been dispatched to countries around the world to help people in need, Washington has also announced tariffs for countries that buy Venezuelan oil.

The potential visa restrictions come as the U.S. is also mulling possible wider-ranging visa bans for some smaller Caribbean nations, under which foreigners can pay for access to visas or passports.

(Reporting by Humeyra Pamuk and Sarah Morland; Additional reporting by Daphne Psaledakis; Editing by Brendan O'Boyle and Aurora Ellis)