The field, known as Arash in Iran and Dorra in Kuwait and Saudi Arabia, falls under "joint ownership between the Kingdom of Saudi Arabia and the State of Kuwait, and they alone have full rights", the official Saudi Press Agency reported, citing a foreign ministry source.
Heavy clashes, shootings, rocket strikes and a car-ramming attack have marred a period when the Muslim holy month of Ramadan coincides with the Jewish Passover and Christian Easter.
"The targeting of innocent civilians of any nationality is unconscionable. The United States stands with the government and people of Israel," State Department spokesman Vedant Patel said in a statement.
After years of hostility that fuelled conflicts across the Middle East, Iran and Saudi Arabia agreed to end their diplomatic rift and re-open embassies in a major deal facilitated by China last month.