The Constitutional Monarchy Movement advocates the return to power of the Hashemite royal family, who currently rule Jordan and who ruled Iraq when the country was under British administration.
Out of three family members who could make claims to the throne, the exiled merchant banker Sharif Ali Bin-al-Husayn has been the most active.
He is the cousin of King Faisal II, who was deposed and killed in a military coup in 1958. Having fled Iraq at the age of two, he returned from London on 10 June 2003.
He is calling for a referendum on the formation of a constitutional monarchy.
The number of Iraqis who actually support the return of the monarchy is probably very few, but there is some speculation that the Hashemites might help bridge sectarian divisions - they are Sunnis but are revered by many Shia as descendants of the Prophet Muhammad.
Another possible contender could be Prince Raad, who describes himself as the "head of the Royal House of Iraq".
Prince Hassan of Jordan, who is related to two former kings of Iraq, King Faisal I and King Faisal II, is seen as an outside candidate – although observers say he may be seeking a role as ambassador to the UN.