Opposition parties have lambasted Erdogan for the delay in even trying to form the government after the June 7 elections, a lag unprecedented in modern Turkish political history.
Erdogan asked Justice and Development Party (AKP) chief Davutoglu to form the new government in talks at his presidential palace in Ankara, the presidency said in a short statement on its website.
The ruling AKP, co-founded by Erdogan and led by Davutoglu, won the most seats in the elections but lost its overall majority for the first time since coming to power in 2002.
The loss of the AKP's majority was seen as a blow for the ruling party as well as the authority of Erdogan himself, who dominated Turkey from 2003-2014 as premier and since August last year as president.
Analysts have seen an AKP-MHP coalition as the most likely option, with an AKP-CHP "grand coalition" also possible. The HDP has ruled out taking part in a coalition with the AKP.
An anti-AKP coalition between the opposition was seen as a non-starter, as the nationalist MHP refuses to work with the HDP.
Davutoglu early today said he expected to conduct initial talks on forming a coalition in the next week before official business in the country winds up on July 17 for the post-Ramadan holidays.
Speaking to his party earlier Thursday, Davutoglu promised to be "constructive" and not to "deprive Turkey of a government".
But in a television interview late yesterday the premier had again refused to rule out early elections if the need arises.