By Jeff Franks

HAVANA (Reuters) - Former President Jimmy Carter will meet with Cuban President Raul Castro, the leader of Cuba's Catholic Church and the island's Jewish community during a Havana visit starting on Monday amid speculation he will seek the release of a jailed American aid contractor.

His public schedule was issued on Saturday by the Cuban government, which invited him for his second visit to the country, the first coming in 2002.

He is the only U.S. president, in or out of power, to come to Cuba since a 1959 revolution that transformed the island into a communist state.

Since leaving office in 1981, Carter has on occasion served as an unofficial diplomatic troubleshooter, including last August when he went to North Korea to secure the release of an American imprisoned there.

The Carter Center in Atlanta, Georgia, said on Friday that Carter, 86, would be in Cuba for three days on a "private, non-governmental mission" to "learn about new economic policies and the upcoming (Communist) Party congress and to discuss ways to improve U.S.-Cuba relations."

The latter have been stymied by the case of Alan Gross, who has been jailed in the Cuban capital since December 3, 2009. Following a trial this month he was sentenced to 15 years in prison.

A Cuban court found that he had committed "acts against the independence and territorial integrity of the state" for attempting to provide illegal Internet access to Cuban groups.

Gross, 61, was in Cuba working under a controversial U.S.-funded program promoting political change on the island, which Cuba views as part of longstanding U.S. efforts to undermine the government.

DESPERATE FAMILY

His wife, Judy Gross, has pleaded for his release on humanitarian grounds because their 26-year-old daughter and his 88-year-old mother have cancer.

On Saturday she welcomed Carter's visit to Cuba and told Reuters in a statement: "If he is able to help Alan in any way while he is there, we will be extraordinarily grateful.

"Our family is desperate for Alan to return home, after nearly 16 months in prison," she said.

The United States, which has said there will be no improvement in U.S.-Cuba relations until Gross is freed, contends that Gross was in Cuba only to set up Internet access for Jewish groups.

Carter's first known stop on Monday will be the main headquarters for Cuban Jews, located in Havana's Vedado neighborhood.

He will then meet with Catholic Cardinal Jaime Ortega, who spearheaded talks with Castro last year that led to the release of dozens of Cuban political prisoners.

On Tuesday Carter will visit a Catholic convent in Old Havana, then meet with President Castro in the afternoon.

It also was likely he would visit the president's older brother, former Cuban leader Fidel Castro, but that has not been announced.

Carter will wrap up on Friday with a press conference before leaving at a still unconfirmed time, the government said.

Carter is respected by Cuban leaders because during his 1977-81 term as president he took significant steps to improve U.S. relations with Cuba, which have been bitter since the revolution.

But the 1980 Mariel boatlift, in which Fidel Castro allowed 125,000 boat people to flee to the United States, hurt Carter politically and contributed to Ronald Reagan defeating him in his quest for a second term.

(Editing by Xavier Briand)