Some of Maliki's opponents, including Barzani and Sadr, have accused him of becoming a dictator and several of his critics say he deliberately sidelines Sunnis and Kurds.
The letter outlines eight demands to be met by May 13 to ensure the four leaders' support for Maliki's administration.
"In case of a refusal to comply with the principles and frameworks of this agreement, practical steps will be taken, within a period of time not exceeding 15 days, to act upon a vote of no confidence against the government," says the letter.
The letter is dated April 28, the last day of a three-day mini-summit held by the leaders during which they said they tried to find a solution to the political impasse.
Maliki did not attend the meeting, which was held in the Kurdish capital Arbil.
The letter called on the government to stop interfering with the security forces and with the work of Parliament and contained a paragraph criticizing autocratic decision-making.
"Putting an end to any kind of one-man decision-making in the government pyramid and a tendency towards autocracy," the sixth demand reads.
The letter also demanded that a two-term limit for the post of prime minister be retroactively introduced. Maliki is serving his second four-year term as prime minister.
"(This is necessary) in order to ensure a peaceful transfer of power and to establish the foundations and principles of democracy and in order not to allow a climate of dictatorship," it said.
The sole Shiite signatory of the letter, Sadr, has a history of acrimonious relations with Maliki.
Maliki's bloc met on Thursday and pledged to hold a meeting of all of the coalition's political blocs within a week to hammer out a solution to the crisis, Ibrahim Al-Jaafari, chairman of the coalition, said in a statement on his website.
Intellpuke: Sadr also has "a history of acrimonious relations" with the U.S.
You can read this Reuters article, filed from Baghdad, Iraq, in context here: arabnews.com/middleeast/article626107.ece