The Iraqi Accordance Front, a Sunni bloc that has 44 seats in the 275-member parliament, has been under pressure from the dominant Shia and Kurdish parties to oust Mahmoud al-Mashhadani because of remarks he made on the Sunni uprising and regional self-rule.
The cross-party consensus to oust al-Mashhadani could result in the first reshuffle of Iraq's fragile government of national unity.
Adnan al-Dulaimi, head of the Iraqi Accordance Front, said he had not heard from al-Mashhadani, a fellow Front member, in several days and did not know if he planned to step aside.
Al-Mashhadani was quoted by The New York Times as saying he was considering resigning.
If al-Mashhadani "is ready to submit his resignation ... this is good and will help a lot in avoiding a crisis in the country", al-Dulaimi said. "He has not talked to us in this regard yet and of course we will agree with the majority opinion."
"In my point of view, the person who killed Americans in defence of his country, in other countries, they would build a statue for him"
Iraqi speaker of parliament
Since taking office in May, al-Mashhadani has spoken out against regional self-rule, strongly supported by Shia and Kurds but opposed by many Sunni Arabs.
He also told reporters last month that if the government refused to grant amnesty to Sunni fighters who killed Americans, "we should punish the American soldiers who killed an Iraqi who fought against occupation".
"In my point of view, the person who killed Americans in defence of his country, in other countries, they would build a statue for him," he said.
Leading politicians from the Shia United Iraqi Alliance also confirmed they want al-Mashhadani to be removed from his position.
Wael Abdul Latif, spokesman for the secular party of the former prime minister, Iyad Allawi, said his group opposed the Sunni politician because he "is disrespectful".
Al-Mashhadani courted further controversy when he claimed "the Jews" were financing acts of violence in Iraq in order to discredit Islamic religious parties that control parliament and the government.
"Beheading, kidnappings and killing... These acts are not the work of Iraqis. I am sure that he who does this is a Jew and the son of a Jew"
Iraqi speaker of parliament
"Some people say, 'We saw you beheading, kidnappings and killing. In the end we even started kidnapping women who are our honour," he said. "These acts are not the work of Iraqis. I am sure that he who does this is a Jew and the son of a Jew."
Al-Mashhadani could be relieved of his duties of speaker via a vote when parliament returns from its summer recess on September 1. His likely replacement would be another Sunni Arab.
A power sharing agreement in the unity government ensures key posts are distributed among Shia, Sunnis and Kurds according to the number of seats they control in parliament.