Even fellow Shiites are saying Maliki, who controls all Iraq's military, security and intelligence forces, should go.
At the same time, Tehran is seeking to ensure that Iraq's Shiites don't upstage Iran's long-held spiritual domination of the Shiite sect, a position that the Iranian clergy seized during Saddam Hussein's rule when he ruthlessly suppressed Iraq's Shiite majority.
The political and religious friction is being intensified by the campaign of bombings and assassinations carried out by remnants of al-Qaida and other militants from the Sunni minority, which many fear will reignite open sectarian warfare in Iraq.
The swelling political crisis that has engulfed the oil-rich state since U.S. forces completed their withdrawal in December 2011 came to a head last week when Maliki's coalition partners moved together to muster support for a no-confidence vote in Parliament.
One of the driving forces behind this campaign is Moqtada al-Sadr, scion of a prominent Shiite religious dynasty and long one of Maliki's rivals for the leadership even though he was persuaded to join the ruling coalition put together, with considerable Iranian effort, in 2011.
Sadr, who fought the Americans after their 2003 invasion and then battled Maliki's forces, has joined forces with minority Sunnis and Kurds to bring down Maliki.
The Iranians have largely sat on the sidelines in recent months, discreetly using the immense influence they've built up in Iraq since the Americans conveniently got rid of Saddam for them.
But now they're concerned the opposition to Maliki is gaining too much strength amid the violence, political paralysis, worsening economic problems, intensifying squabbles over oil and revenue-sharing and the fear among Iraqis that Maliki' determined to become another Saddam Hussein.
Sadr, a cleric who spent four years studying in the Shiite holy city of Qom, south of Tehran, to avoid being nabbed by the Americans, was summoned to Tehran for talks Monday. Iraqi sources said it was clear the Iranian leadership is seeking to get Sadr to back off.
Tehran fears that if Maliki's pushed out, sectarian bloodshed will ensue.
On the other hand, if Maliki does stay in power, there's a growing belief the Kurds, who already have a semiautonomous enclave in the north, will break away altogether and that restive Sunni regions and even Shiites in the oil-rich south will follow suit.
The Iranians have already leaned on Jalal Talabani, Iraq's president and a powerful Kurdish leader, to back off the no-confidence vote.
Talabani, who often allied with Iran during the 1980s and '90s in the Kurds' secessionist war against Baghdad and is a close U.S. ally, has been meeting political factions who want to be rid of Maliki.
So far he hasn't pushed too hard on the no-confidence vote and the Iranians have reportedly pressed him not to escalate the crisis by putting his considerable political influence behind such a move.
It's not yet known what Tehran may be offering as an alternative solution.
But it's clear Tehran's alarmed at the prospect of its efforts to dominate Iraq, a longtime foe, falling apart at the same time that its key Arab ally, the Syrian regime of President Bashar Assad, is under threat from a 15-month-old pro-democracy uprising.
The loss of influence in these two states on Iran's eastern flank, the Islamic Republic's gateway to the Levant and Israel's very doorstep, is simply not acceptable.
So it's odd to find that while U.S. President Barack Obama's administration wants to see Assad toppled and Syria cut loose from Iran, it doesn't want to see Maliki pushed out and Iraq plunged back into political turmoil and sectarian warfare.
That gives greater weight to Iran's efforts to ensure that an Iranian ayatollah takes over as leader of Iraq's Shiites when the moderate incumbent, Iraq's 81-year-old Grand Ayatollah Ali al-Sistani, steps down.
Sistani is the marja -- spiritual leader -- of all Shiite Muslims and he wields enormous power over the sect.
But most importantly for Tehran, he also rejects the concept of religious rule in Iran.
If an Iranian ayatollah takes his place, Tehran would find its plan to dominate Iraq much easier.