The Iraqi Islamic Party, Mr Allawi's main Sunni Arab partner, said it would leave the government.
The European Union and Nato alliance both pledged their support for Mr Allawi's government.
If Mr Allawi follows this lead, Iraq could end up in a vicious circle.
Allawi also expressed gratitude for the recent NATO decision to help train the Iraqi military.
The announcement came after a meeting of Iraqiyya's senior figures, including former Prime Minister Iyad Allawi.
BBC: Iraqi bloc ends parliament boycott over Hashemi warrant
But even before his swearing-in on June 30th, Mr Allawi has flexed his muscles.
"President Allawi will have an answer on what he has asked, " de Hoop Scheffer said Saturday.
It has singled out 52 candidates in the recent election, including 22 from Mr Allawi's slate.
Mr Allawi also got a warm reception during a recent tour of eight Arab capitals.
Maybe, say Mr Allawi's people, such areas can simply be fenced off and rendered voteless.
Plainly there is no light yet at the end of Mr Allawi's tunnel to normality.
The Rose Garden appearance followed an address Thursday morning by Allawi before a joint meeting of Congress.
Allawi is considering a state of emergency to enforce law in Iraq after the June 30 handover.
Mr Allawi calmly and methodically deconstructs an Islamic revival which has failed to live up to its promise.
In Ireland on Saturday for a summit with European Union leaders, Bush urged NATO to accept Allawi's request.
Still, Mr Talabani's Kurds may want Mr Allawi in to act as a counterweight to the UIA's pushier elements.
ECONOMIST: Iraq's Shias stick with the devil they know | The
Rival alliances led by the prime minister, Nuri al-Maliki, and by Iyad Allawi, a more secular predecessor, scored well.
Powell expressed confidence Sunday that NATO will give a positive response to Allawi's request for assistance in training troops.
Hence Mr Allawi is trying his hardest to accommodate the Sadrists there without simply handing over power to them.
"It's going to be a very transparent and very just trial, " Allawi said.
Iyad Allawi, the prime minister, also said the death penalty may come back.
Zelikow, who advises the firm on other issues, told CNN he has never been asked by the firm about Allawi.
Mr Allawi argues that, provided the Americans do not cut and run, the insurgency can be contained, if not beaten.
Allawi was interviewed in London during a trip to meet with British Prime Minister Tony Blair earlier in the day.
If Mr Allawi, with armed multinational help, can provide the former, the latter may, with a lot of luck, follow.
Last weekend, Mr Allawi declared a 60-day state of emergency in all parts of Iraq except the relatively tranquil Kurdish-majority north.
In the true spirit of a Coalition Government, Allawi, accepted the powerful position of President of the newly formed National Council.
FORBES: Is The Iraqi Dinar Worthless Paper Or Maker Of Millionaires?
Those elections gave the Sunni-dominated Iraqiya party led by former prime minister Ayad Allawi the right to form the next government.
CENTERFORSECURITYPOLICY: Calling things by their proper names
Allawi insisted Iraqi elections would be held in January, as planned, but said he had not decided whether to seek office.
But the latest row concerns the Kurds, who backed Mr Allawi's selection and have a strong showing in the new government.
应用推荐