However the move was blocked by US Ambassador to the UN Nikki Haley.
Some UN officials believe Mr Guterres offered Ms Livni the under-secretary general post as part of an attempt to persuade the Americans to allow the appointment of Mr Fayyad in the Libya role.
On Sunday, Israeli Prime Minister Benjamin Netanyahu said at a cabinet meeting he had been told about the Fayyad move several days ago, and had pushed for a quid pro quo that would be fair to Israel if the appointment went ahead.
Mr Netanyahu said: “I said that the time has come for parity in the attitude toward Israel, and that the Palestinian side can't be given freebies all the time. The time has come for the Israeli side to get status and appointments if [Fayyad] is tapped for the position of special UN envoy to Libya.”
