Hussein al-Radi was detained after Gulf Arab states declared Hezbollah a "terrorist" group earlier this month and brought in tough new measures against anyone supporting it.
The Al-Watan daily reported that security forces arrested Radi, from the Al-Ahsa oasis region in Eastern Province.
"This is after he glorified the terrorist group Hezbollah and insulted the kingdom in a video clip that has been shared" online, the report said.
Radi "also broke previous pledges he had made after defending the terrorist Nimr al-Nimr following his execution," it added.
Nimr, another Shiite cleric from Eastern Province, was a driving force behind protests that began in 2011 among the Shiite minority in Sunni-majority Saudi Arabia.
The Shiites complain of marginalisation.
Nimr and three other Shiites were among 47 people executed on January 2 for "terrorism."
Iranian demonstrators stormed the Saudi embassy in Tehran and a consulate following Nimr's execution, prompting Riyadh to cut diplomatic ties.
Al-Watan quoted a security source as saying authorities had shown "patience" over a number of violations Radi allegedly made.
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"But he continued to incite the public, taking advantage of the mosque platform to breach regulations."
A video posted Sunday on YouTube showed the bespectacled Radi, with a bushy white beard, speaking at a podium where he hails Hezbollah leader Hassan Nasrallah as a "hero."
Radi also praises Iran as a regional and international power.
An interior ministry spokesman could not confirm Radi's arrest to AFP but said: "I would assure you that ... laws in the kingdom are enforced."
Ibrahim AlMugaiteeb, president of the Eastern Province-based Human Rights First Society, said Radi's arrest was no surprise.
But even though the imam "pushed the envelope," AlMugaiteeb said he did not condone the arrest of an elderly religious figure.
Other Gulf states have also taken measures against alleged Hezbollah supporters since the "terrorism" blacklisting.
Bahrain announced it had deported several Lebanese residents for alleged links to the group.
A Kuwaiti newspaper reported on Monday that authorities there had taken similar action against 11 Lebanese and three Iraqis.
And the United Arab Emirates has reportedly put seven people on trial for allegedly forming a cell linked to Hezbollah.