Allawi's appointment came just hours before a deadline set by Salih for the Iraqi political blocs to nominate a candidate for premiership, amid months of political row and anti-government demonstrations, Xinhua news agency reported on Saturday.
Allawi is an Iraqi politician who served as the minister of communications.
In an online statement, Allawi called on protesters to continue their rallies against corruption and pledged to quit if the political blocs insist on naming ministers for his cabinet.
"If it wasn't for your sacrifices and courage there wouldn't have been any change in the country," he said.
"I have faith in you and ask you to continue with the protests," he added.
Salih on Thursday warned the political blocs that if they fail to nominate a candidate by Saturday, he would use his constitutional powers to assign whoever he finds acceptable by the demonstrators.
The Iraqi parliament is expected to vote on Allawi in the next session.
According to the constitution, the largest coalition in parliament should nominate a candidate for the vacant post to the Iraqi president who will task the new prime minister-designate with forming a new cabinet.
The prime minister-designate has 30 days to form a cabinet and present it to parliament for approval. The parliament must approve each minister in separate absolute majority votes.
Mass anti-government demonstrations have been continuing in the capital Baghdad and other cities in central and southern Iraq since October of 2019, demanding comprehensive reform, fight against corruption, better public services and more job opportunities.
( With inputs from IANS )