"The US forces do not have the right to interfere in a [judicial] case and decide whether it was legal or not," Ali al-Dabbagh said. "Only Iraqi authorities have the right to do so."
The US embassy in Baghdad had no comment on the letter.
US and Iraqi officials have had disagreements over who has the authority to transfer the men for execution.
The three were convicted of genocide for their roles in a campaign against Iraq's Kurds in 1988, but the US military has said it will not hand them over until it receives what it calls an "authoritative government of Iraq request".
What constitutes such a request is at the centre of a row between al-Maliki, Jalal Talabani, Iraq's Kurdish president, and Tareq al-Hashemi, the Sunni Arab vice-president.
Al-Maliki has said the US embassy has played an "unfortunate role" in preventing the handover of the three men.
Talabani and al-Hashemi say Iraq's constitution stipulates that the three-man presidency council - made up of the president and two vice-presidents - should sign the order.
Al-Maliki's government says the council has no such right.
His letter to Bush indicates his government's impatience to carry out the sentence, which was upheld by an Iraqi court in September.
Under Iraq's constitution, the death sentence should have been carried out within 30 days.
Al-Hashemi has threatened to resign if the government goes ahead with the executions without a presidential decree. He said he narrowly stopped the government from executing them in September.
The legal dispute has been complicated by a growing chorus of calls for Hashem's life to be spared. Many say he was a soldier simply following the orders of "Chemical Ali".
Saddam Hussein was hanged on December 30 after being found guilty in a court of law.