A political leader in Beirut told the Daily Star on condition of anonymity Monday the prime minister-designate gave Michel Aoun, the leader of the Hezbollah-allied Free Patriotic Movement, a "take-it-or-leave-it offer" on a new government.
Aoun refused to accept Salam's offer for the Foreign Affairs Ministry or Public Works Ministry if the FPM leader gave up his party's ambitions for the Energy Ministry, the report said.
The Daily Star said Salam plans to issue his list of Cabinet positions later this week.
The report said Hezbollah and its allies may quit the government if Salam goes ahead with his list.
Prime Minister Najib Mikati, a Hezbollah ally, resigned in March because of differences between the pro-western March 14 coalition and the March 8 slate, which includes Hezbollah and Aoun's party.
Salam, son of former Prime Minister Saeb Salam, won support from the pro-Western March 14 alliance and March 8 when he was appointed in April.
Salam was quoted as saying during the weekend the security situation in Lebanon "demands" a resolution to the political stalemate as soon as possible.
At least one person was killed Monday in a suicide bombing in a Hezbollah stronghold of Beirut, the Lebanese newspaper reported.