By Lesley Wroughton

WASHINGTON (Reuters) - The United States and Venezuela have embarked on their most extensive dialogue in years in an attempt to improve their acrimonious relations, according to a senior U.S. administration official.

The quiet diplomacy, the extent of which has not been previously reported, is a sign that U.S. detente with Communist Cuba may be helping to reshape another troubled Latin American relationship. The official, who has direct knowledge of the high-level talks, cautioned that the process is at an early stage.

The effort by Latin America's most ardently anti-Washington government and major U.S. oil supplier to improve relations comes as President Nicolas Maduro struggles with a decaying state-led economy that has been left more isolated by close ally Cuba's warming U.S. ties.

Maduro made the first move in March - around three months after Washington and Havana announced on Dec. 17 they were seeking to restore diplomatic ties - by requesting a "direct channel of communication" with U.S. President Barack Obama and the State Department, said the official. Cuba and the United States announced on Wednesday they had agreed to restore ties.

"He realized that if we can talk to the Cubans, we can talk to him," the official told Reuters, adding: "We approached it very carefully because we had seen this before, but there was also U.S. concern that the relationship was reaching such a dangerous point that it risked breaking completely."

Venezuela's Foreign Ministry, presidency and national assembly did not respond to requests by Reuters for comment on the dialogue with Washington.

Venezuelan officials say they are seeking to repair relations, which hit a nadir in March when Venezuela's socialist government ordered the United States to slash its embassy staff in Caracas and Washington imposed sanctions on top Venezuelan officials.

The two foes have toned down their confrontational rhetoric in recent weeks.

In a further sign of the improving atmosphere, Republican Senator Bob Corker, chairman of the Senate Foreign Relations Committee, is in Venezuela this week to meet both opposition and senior government figures as part of a fact-finding mission.

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TWO-TRACK APPROACH

The dialogue has developed into a two-track effort to separate areas of disagreement, such as Venezuela's clamp-down on domestic political opposition, from those of shared interest including peace talks in Colombia and elections in Haiti, the U.S. official said.

Relations between the two countries have waxed and waned for over a decade, with periods of squabbling and diplomatic expulsions often followed by short-lived periods of conciliatory words.

Venezuela's powerful parliamentary chief Diosdado Cabello has been involved in the talks even though he is a controversial figure, the official said. U.S. media in May reported that U.S. prosecutors were investigating Cabello for possible drug trafficking and money laundering. He denies the allegations.

During a June 14 meeting in Haiti, U.S. officials pressed Cabello to set a date for parliamentary elections this year and to release political prisoners, including jailed opposition leader Leopoldo Lopez, the U.S. official said.

Washington made clear that if Lopez died from his hunger strike, which was partly in demand for an election date, it would signal the end of the dialogue.

The official said Cabello did not commit to setting an election date during the meeting. But a week later Venezuela's election authority, which had seemed to be stalling on a date for polls widely expected to be won by the opposition, announced they would be held on Dec. 6. Lopez ended his protest shortly thereafter, but remains in jail.

"We were very focused on keeping (Lopez) alive, and that meant engaging with the Venezuelans and telling them that Lopez's death would end our rapprochement efforts," the official said.

The official, who asked not to be identified because he is not authorized to speak publicly on the subject, said it was important to engage Cabello because he is seen as a rival for power with Maduro, although the two deny it.

"The two most apparent power centers in the Venezuelan government are Maduro and Cabello," the U.S. official said. "We knew we had to connect Cabello and Maduro in some way. Even though they insist they aren't, they are competitors."

The meeting in Port-au-Prince between Cabello and Tom Shannon, a seasoned diplomat and the State Department's legal counselor, was to seal what the U.S. official described as an early success of the dialogue - a Venezuelan commitment to contribute funding for the election process in Haiti.

Venezuela agreed to provide funding for security and logistics of a U.N.-coordinated program in Haiti for the August polls, the official said, after years of running separate multi-million-dollar aid programs. It also agreed to cooperate in Haiti in the areas of health, energy and agriculture.

ELECTION MONITORING

Cabello was quoted by state-funded network Telesur after the meeting as saying that Venezuela wants better relations with the United States but that unnamed "interests" were blocking that.

U.S. diplomats are also seeking to boost Caracas' role in pushing Colombia's Marxist FARC rebels to lay down their arms as part of ongoing peace talks.

The dialogue with Venezuela did not begin in earnest until April, when Obama briefly met with Maduro in Panama at the Summit of the Americas, according to the official.

Maduro had threatened to confront Obama at the summit over the new U.S. sanctions. To avoid an awkward incident, Shannon traveled to Caracas beforehand to meet Maduro and open channels for dialogue.

U.S. officials have pressured Venezuela to release opposition activists arrested in violent anti-government street protests last year. Rights Group Penal Forum says 75 people are currently imprisoned for political reasons, down from 77 after two were freed last week.

"It is our hope and our effort to see more of these students released," the official said.

The challenge ahead, the official said, was to reach agreement on meaningful observation of the December vote.

The head of Venezuela's elections council said in April that South American regional bloc Unasur would "accompany" the parliamentary vote. The 35-member Organization of American States has said that, if invited, it would be willing to help monitor the elections.

"We figure if we can help Venezuelans construct an election that everybody supports, and if we can help them get a result that everyone recognizes that is a starting point," the official added.

The official said it was too soon to expect relations with Venezuela to normalize anytime soon, including the appointment of ambassadors, or to consider easing sanctions.

"We're not getting back there anytime soon," the official said.

(Additional reporting by Brian Ellsworh and Girish Gupta in Caracas; Editing by Stuart Grudgings.)