Adds graphic, details on imports and exports throughout

By Marianna Parraga and Mircely Guanipa

July 1 (Reuters) - Venezuela's oil exports fell slightly to 1.2 million barrels per day last month from 1.24 million bpd in May as two deadly earthquakes caused minor delays at terminals operated by state-run energy company PDVSA, according to data based on tanker monitoring and company documents.

PDVSA's vast infrastructure was largely untouched by the twin quakes, which have left almost 2,000 people dead and hundreds of buildings affected, official figures showed. But power supply remains an issue to resolve so all of the country's oil processing plants, including the small El Palito refinery and the Moron Petrochemical Complex, can resume operations.

The June average marked the first month of decline after three consecutive months of growth fueled by a flagship oil supply pact between Caracas and Washington agreed to in the aftermath of President Nicolas Maduro's capture in January.

The state company's main oil terminal of Jose in the country's eastern region did not suffer long delays due to the quakes, while heavy crude stocks there remained close to May levels, the documents showed.

Oil minister Paula Henao said last week that fuel inventories and output were enough to meet demand.

However, the condition of oil infrastructure located near the quakes' epicenter, including PDVSA's Catia la Mar domestic fuel terminal, has not been officially reported.

PDVSA did not immediately reply to a request for comment.

In June, shipments to the U.S. rose to almost 630,000 bpd from 558,000 bpd the previous month, while exports to India fell to 277,000 bpd from 427,000 bpd and shipments to Europe declined to 99,400 bpd from 169,000 bpd. A larger number of Venezuelan oil and fuel cargoes were stored in Caribbean terminals, according to the documents and data.

Chevron's CVX.N exports of Venezuelan crude rose to some 293,000 bpd in June from 268,000 bpd in May, while trading firms including Vitol and Trafigura exported some 775,000 bpd, slightly below the 787,000 bpd of the previous month.

U.S. refiner Phillips 66 PSX.N, which earlier this year agreed to a contract to buy oil and fuel directly from PDVSA, received at least two cargoes bound for the U.S. in June, while trading firm Novum Energy is loading a cargo for July delivery, the documents showed.

At the Orinoco Belt, the country's largest crude output region, the Petromonagas project restarted a delayed coker in mid-June with an initial 30,000 bpd of processing, one of the documents showed.

Venezuela also exported 176,000 metric tons of oil byproducts and petrochemicals last month, below the 240,000 tons shipped in May. The country's imports of naphtha rose to some 106,000 bpd from 93,000 bpd in May.