Seyyed Ahmad Khomeini, a descendant of Ayatollah Ruhollah Khomeini, told The Times that the legacy of his family name was at stake in the conflict.
They rose more than 4 percent as traders wondered if the United States would take a more active role in the conflict between Israel and Iran.
American commanders have put troops on high alert throughout the region as fears of a wider war grow.
A bipartisan pair in the House and a Democrat in the Senate have moved to force votes in the coming days on requiring congressional approval before U.S. troops could take offensive action against Iran.
The cleric has ruled for more than three decades, spearheading a policy of backing proxy groups to wield influence across the Middle East.
Israel’s military campaign reminds some of America’s ill-fated Middle East interventions, which President Trump has long denounced.
Hezbollah, Iran’s most powerful regional ally, has signaled privately that it does not intend to attack Israel, but Lebanon’s military has bolstered its presence in the country’s south to prevent rocket fire.
Since shortly after it was founded in 1948, Israel has been intent on building a nuclear program to ensure its survival.
President Trump spent the first months of his term holding back Israel’s push for an assault on Iran’s nuclear program. With the war underway, his posture has gyrated as he weighs sending in the U.S. military.
Analysts say the Kremlin is prioritizing its own war against Ukraine, as well as its relations with Gulf nations that don’t want to see a stronger Iran.