The king, who is breathing with the help of a tube due to pneumonia, is in hospital in Riyadh, prompting frenzied debate about the future of the oil-rich Gulf state.
Iranian newspapers are running daily reports on his health, partly explained by an often tense relationship -- most recently over falling world oil prices -- between the Sunni kingdom and Tehran's Shiite regime.
Vatan-e-Emrooz, a conservative daily, has been summoned over a front page story on January 6 headlined with a Persian expression often interpreted as "may I hear the news of his death".
The official IRNA news agency said Vatan-e-Emrooz was summoned because its report may have "violated national interests," but it did not say when a court hearing would take place.
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The story also carried sub-headings, including; "Signs of collapse of Saudi Arabia," "Death of Abdullah and USD 200 oil," "Wahabism equals terrorism," and "Crisis of succession of Abdullah."
Saudi Arabia is the world's largest oil exporter and the biggest producer in the Organisation of the Petroleum Exporting Countries, to which Iran also belongs.
In a letter to Iran's media regulator, the Vatan-e-Emrooz editor defended the Abdullah report, saying his health is the focus of unprecedented attention, according to Mehr news agency.
While Riyadh says it is financially strong enough to withstand the drop in oil prices by around 50 per cent in the past year, the budgets of Iran and other oil sales reliant nations are under strain.
The Iran-Saudi relationship deteriorated when they took opposite sides on the war in Syria, with Riyadh supporting rebels while Tehran backs President Bashar al-Assad's regime.