"Soft coup in Saudi Arabia/Son becomes the succesor of the father," read the headline on the Iranian state TV website.
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The move is likely to rattle Iran's leadership, which has been critical of comments by Prince Mohammed last month that the "battle" should be taken into Iran.
Iran, which is predominantly Shi'ite Muslim, and Saudi Arabia, which is mostly Sunni, compete for power and influence across the region. The two countries support opposite sides in the conflicts in Syria and Yemen.
Islamic State carried out attacks on the Iranian parliament in Tehran and the shrine of the founder of the Islamic Republic, Ayatollah Ruhollah Khomeini, two weeks ago that left at least 17 dead. Iranian officials pointed a finger at Saudi Arabia, which denied any involement.
Tension between the two countries increased last month when U.S. President Donald Trump visited Saudi Arabia and took a bellicose stance against Iran in an official speech, repeatedly accusing the Islamic Republic of supporting terrorism.
Iranian Supreme Leader Ayatollah Ali Khamenei labeled the Saudi leadership "idiots" in a speech last month.