(Adds phone call with Saudi king)

DUBAI, Sept 15 (Reuters) - Kuwait is investigating the sighting of a drone over its territory and coordinating with Saudi Arabia and other countries after Saturday's attacks on Saudi oil plants, the cabinet said on Sunday.

"The security leadership has started the necessary investigations over the sighting of a drone over the coastline of Kuwait City and what measures were taken to confront it," the cabinet said on Twitter account.

It said Prime Minister Sheikh Jaber al-Mubarak al-Sabah directed military and security officers to tighten security at vital installations in the Gulf OPEC producer and to take all necessary measures "to protect Kuwait's security".

Some Iraqi media outlets have said Saturday's attack on Saudi oil facilities came from Iraq, which borders Kuwait. But Baghdad denied this on Sunday and vowed to punish anyone using Iraq, where Iran-backed paramilitary groups wield increasing power, as a launch pad for attacks in the region.

Kuwait's 90-year-old ruler Sheikh Sabah Al-Ahmad Al-Jaber al-Sabah, who left a U.S. hospital last week after completing medical tests, spoke with Saudi King Salman to express support for the kingdom following the attack, state news agency KUNA reported. (Reporting by Hadeel Al Sayegh and Aziz El Yaakoubi in Dubai; Additional reporting by Ahmed Hagagy in Kuwait; Writing by Ghaida Ghantous; editing by Emelia Sithole-Matarise, William Maclean)

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