The adviser added that al-Maliki wants to change the Iraqi constitution before he leaves office to limit all future prime ministers to two terms.
If he decided to run again, he would be seeking his third term.
Al-Maliki's successor will be the first to lead the country without U.S. military help since the fall of Saddam Hussein.
American troops plan to fully withdraw from Iraq by the end of 2011.
Saturday's stunning announcement follows al-Maliki's decision a day earlier to return half of his annual salary to the government - a move he said was aimed at narrowing the wide gap between rich and poor Iraqis.
Al-Maliki is not required to publicly report his pay, but he is believed to earn at least $360,000 annually. The U.S. government estimates that as many as 30 percent of Iraqis are unemployed.
The Associated Press contributed to this report.