Protesters have threatened to shut down the facilities, with one group campaigning against corruption issuing a 72-hour ultimatum that ended on Friday.
The two facilities are the Mellitah complex and the Zawiya refinery.
Mellitah is a joint venture between Libya's National Oil Corporation (NOC) and Italy's Eni. If the complex is closed, that would disrupt the supply of gas through the Greenstream pipeline between Libya and Italy.
Salem Mohamed, the spokesman for the protesters, a group called the Corruption Eradication Movement, said that they decided to extend the deadline to continue negotiations with the six-person mediation team.
"There was a consensus during the negotiations on our demands, except one point - the dismissal of NOC chairman Farahat Bengdara," Mohamed added.
"If they do not return to us with an agreement on our all demands, especially the dismissal of Bengdara and the cancellation of all his decisions, we will close the Mellita complex and Zawiya refinery on Saturday afternoon."
It was unclear whether the protesters have the capacity to close the two facilities.
Libya's oil sector, the country's major source of income, has been a target for local and broader political protests since the toppling of Muammar Gaddafi in a NATO-backed uprising in 2011.
The Zawiya refinery, with a capacity of 120,000 barrels per day (bpd), is connected to the country's 3000,000 bpd Sharara oilfield.
Sharara was closed by protesters from the Fezzan region in the south last week to demand "better services" for the region.
That had forced NOC to declare force majeure in the field production and suspended the supply of crude oil to Zawiya terminal, NOC said.
(Reporting by Reuters Libya newsroom, writing by Yomna Ehab, editing by Chris Reese and Louise Heavens)
