Cruise giant Carnival , under fire for perceived discrimination, reversed course Monday and started accepting reservations for its planned trips to Cuba from people born on the island.

The US leisure group plans to start cruises from the United States to Cuba on May 1 as part of the historic restoration of diplomatic ties between the two countries.

But under a rule dating back to the Cold War, the Cuban government only allows Cubans to enter the country by air, not by ship.

Carnival said it is holding talks with the Havana government and trusts this restriction will be lifted.

If it is not, the May 1 launch of the cruise trips by a new Carnival unit called Fathom will be postponed, the company said in a statement.

"Carnival Corporation (NYSE: CCL - news) continues active discussions with Cuba, asking that travel on Fathom be on a level playing field with air charter travel to Cuba and remains confident its discussions with Cuba will result in a positive outcome for everyone who wants to travel to Cuba, including those who are Cuba-born," Carnival said in a statement.

Carnival is the first cruise ship company to receive permission from the United States and Cuba to offer trips between the two countries for the first time since they were halted with the Cuba Revolution in 1959. The bilateral rapprochement began in 2014.

But Carnival came in for criticism by not accepting reservations from Cuban-born people, in line with the current Cuban law. The uproar came from anti-Castro activists in Florida, US lawmakers and even the government of President Barack Obama.

"Carnival needs to not discriminate," Secretary of State John Kerry said last week in an interview with the Miami Herald, although his ire was really aimed at the Cuban government.

"The United States government will never support, never condone discrimination. And the Cuban government should not have the right to enforce on us a policy of discrimination against people who have the right to travel," Kerry said.