The telephone talks come a day after GNA chief Fayez al-Sarraj met with the US Ambassador to Ankara David Satterfield in Istanbul.
After talks in Moscow aimed at halting Haftar's offensive on the capital failed to yield a binding ceasefire agreement on Monday, the Turkish president said Ankara would not refrain from "teaching a lesson" to the LNA if its attacks continue.
Libya's Khalifa Haftar, commander of eastern Libyan forces, arrived in Athens Thursday and was expected to have talks there on Friday with Greek Prime Minister Kyriakos Mitsotakis and Foreign Minister Nikos Dendias during a stopover on his way to Berlin for a conference on the Libya crisis.
Haftar "has agreed to abide by the ongoing ceasefire", he added. Haftar left Moscow without signing the proposal.
Libya has been fractured and deeply unstable, with outside powers providing support to rival armed factions since Muammar Gaddafi was toppled in a 2011 uprising, which prompted the arms embargo.
Testa stressed the need to withdraw all foreign forces that have fueled the conflict in the country, urging all Libyan parties to seize this opportunity to address the political, economic and security issues that divide them.
The committee also expressed support to the political leadership in all measures it perceives as mandated to deal with the situation.
With an worldwide conference on the Libyan conflict due in Berlin on Sunday, Akar said Turkey supported "a ceasefire, peace, a political solution".
Russian Federation and Turkey have turned into important players in Libya, joining Arab countries such as Egypt or the UAE which have filled a void left by Western powers showing little interest in the OPEC producer since 2011.
Gen Haftar is also backed by Egypt, Saudi Arabia, the United Arab Emirates (UAE), and Jordan, raising fears that oil-rich Libya could become the theatre of a regional conflict, or even a "second Syria".
According to a draft of the deal which Sarraj signed and which was shared by a GNA spokesperson with Middle East Eye, both sides would create a joint military commission to ensure the stability of the truce. Several hundred Russian mercenaries are reported to be in Libya supporting Haftar.
Tunisia, which shares a 450-kilometer (280-mile) border with western Libya and fears an escalation could spark an influx of thousands of refugees, voiced disappointment Thursday at not being invited to the conference. "This meeting is an opportunity for the U.K.to come together with global partners and countries in the region to support efforts towards a cease-fire and encourage a return to United Nations-led talks".
Tunisia's ambassador to Germany conveyed his consternation in an interview with Deutsche Welle, his sentiments confirmed by Tunisia's foreign ministry, which would not comment further. German Foreign Minister Heiko Maas arrived in Benghazi on Thursday on a mission to persuade him.
What is the Libya conflict?
The fighting has spurred a growing exodus of migrants, many embarking on rickety boats towards Italy.