Political tensions remained on Tuesday, as the coalition of parliament speaker Nabih Berri announced it would not name anyone for the post.
Hariri was still expected to gain enough support to be renamed prime minister-designate.
Leader of the March 14 alliance, Hariri was appointed on June 27 after he and his allies won a majority of 71 seats in an election earlier that month.
The rival Hezbollah opposition and its allies won 57.
Hariri has said that the conditions set by the Hezbollah-led bloc in talks to form a unity cabinet sought to undermine the election.
Rival alliances had agreed on the broad division of seats in the cabinet, but could not agree on the details.
Under Hariri's proposal, his alliance would have had 15 seats, the Hezbollah-led minority would have had 10, and Suleiman would pick the remaining five.
However, the opposition dismissed the make up and put forward an alternative.