"The first objective for mediators is (...) to try and get the federal and Tigray delegations to agree to a truce despite the momentum towards continued military confrontation," said William Davison, senior Ethiopia analyst at the International Crisis Group think tank.
Humanitarian access and restoration of basic services to Tigray and its six million people is another key demand.
The Pretoria negotiations are set to run until Sunday -- but what's next is unknown and analysts say an immediate breakthrough is unlikely.
"Battlefield developments have a direct bearing on power dynamics at the negotiating table and threaten to overtake political discussions given the current momentum of pro-government forces operating in Tigray region," the Eurasia Group said in a note.
It said Addis Ababa appeared to be ratcheting up its demands, noting Demeke had called for the "dissolution of Tigrayan forces" before any settlement.
"Addis will now aim to leverage its military ascendancy to force a TPLF climbdown from red lines on disputed territories and restoration of basic services which have thus far thwarted talks."
Fighting resumed on August 24, torpedoeing a five-month truce that had allowed limited aid into war-stricken Tigray, with both sides accusing the other of firing first.
Since then, pro-government forces have gained ground in offensives Addis Ababa says are aimed at protecting Ethiopia's "sovereignty and territorial integrity" and taking control of airports and other federal sites in Tigray.
Ethiopian and Eritrean forces, using artillery bombardments and drone strikes, have captured a string of towns in Tigray including the strategic city of Shire, driving toward the capital, Mekele.
But Tigrayan leader Debretsion Gebremichael remained defiant, saying victory was "inevitable."
"We're going to South Africa while still fighting," he said early this week. "Those joint enemy forces that entered Tigray will be buried."
The TPLF and the international community also want the withdrawal of troops from Eritrea, a feared and unpredictable player in the conflict.
Its soldiers, whose presence in the first phase of the war had long been denied by Addis Ababa, have been accused of atrocities against civilians.
Humanitarian access and restoration of basic services to Tigray and its six million people is another key demand by Tigrayans and, larger, the world community.
The region, where many are going hungry, is largely cut off from the outside world, with a communications blackout and shortages of food, fuel and medicines.
Addis Ababa said this week it was working with humanitarian agencies to provide aid in the areas it had taken over.
