The move could end the paralysis gripping Iraq's political elite, which has struggled to respond to four months of anti-regime rallies in Baghdad and mainly-Shia southern Iraq.
In a video he posted to Twitter, Allawi, 65, said President Barham Saleh had nominated him to form a new government and that he would do so in line with protesters' demands.
Faced with the stalemate, Saleh had told Iraq's divided parliament that he would name his own candidate unless it nominated someone by Saturday.
"After the President appointed me to form a new government a short while ago, I wanted to talk to you first," Allawi said, addressing the camera in Iraqi dialect.
"I will ask you to keep up the protests, because if you are not with me, I won't be able to do anything," Allawi said.
There was no formal announcement from Saleh or other government bodies.
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Under the constitution, Allawi now has one month to form a cabinet, which would require a vote of confidence from parliament.
In Iraq, the cabinet is typically formed by consensus among political rivals after intense horsetrading over influential posts.
"If the (political) blocs try to impose their candidates on me, I'll come out and talk to you and leave this nomination," Allawi said.
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