By Humeyra Pamuk and Kemol King

GEORGETOWN, Guyana, March 27 (Reuters) - It would be "a

very bad day" for Venezuela if it were to attack its neighbor

Guyana or U.S.-based energy giant ExxonMobil ( XOM ), U.S.

Secretary of State Marco Rubio said on Thursday during a visit

to Guyana's capital.

Guyana and Venezuela are locked in a long-running dispute

about which country has rights over the 160,000-square-km

(62,000-square-mile) Esequibo area, which is the subject of an

ongoing case at the International Court of Justice.

Washington has offered military support to the tiny South

American country amid the territorial dispute and increasing

U.S. sanctions on Venezuela.

The U.S. Navy cruiser Normandy and the Guyana Defence

Force patrol vessel Shahoud were conducting exercises in

international waters and the Guyana Exclusive Economic Zone, the

U.S. embassy in Guyana said in a social media post early on

Thursday.

Tensions rose early this month when Guyana said a Venezuela

coast guard patrol entered its waters and approached an output

vessel in an offshore oil block operated by Exxon.

A consortium by Exxon, Hess and China's CNOOC

controls all oil and gas output in Guyana, which this year is

producing some 650,000 barrels per day.

The northwest portion of the block, close to Venezuela,

has remained in force majeure as the Exxon group has been unable

to complete exploration there.

Venezuela's communications ministry did not immediately

respond to a request for comment.

Venezuela has previously said the vessel did not enter

Guyanese waters as the maritime zone delimitation is still

pending as part of the territorial dispute.