Fighting quickly took on an ethnic nature, pitting Kiir's Dinka tribe against Machar's Nuer.
Machar yesterday told the French Press Agency that his forces were marching on the capital Juba.
The rebel leader claims that Nuer militia under his command on Tuesday captured Bor, the capital of Jonglei state, just 200 kilometres north of Juba. South Sudanese army spokesman Philip Aguer disputed the claim that Bor had been seized, saying fighting was ongoing inside the town.
President Kiir has described the war as "senseless", but has ruled out any power-sharing arrangement with the rebels.
South Sudan also dominates front pages in Uganda, with the main story in the Kampala-based Daily Monitor headlined "Machar accuses Uganda of fuelling South Sudan conflict".
South Sudan's former vice president Riek Machar has responded to Uganda's President Yoweri Museveni, accusing Uganda of fuelling the fighting in the world's youngest nation. "We call upon the African Union and the Inter-Governmental Authority on Development to restrain the Ugandan government from fuelling the conflict by sending troops and war planes in support of the government of Salva Kiir," a statement released to media said.
In Juba on Monday, Museveni warned Dr Machar to either agree to the ceasefire or face collective military action from IGAD member states. The rebels now say they are willing to talk, observing in their statement carried by the Sudan Tribune newspaper yesterday that: "We are ready to ceasefire immediately to stop the bloodletting once the government of Salva Kiir reciprocates."
The main story in regional paper The East Africa says that tourists visiting Kenya, Rwanda and Uganda will no longer have to pay for separate visas for each of the three countries, as the East African single tourist visa comes into effect today, January 1, 2014.
Working like the Schengen visa for European Union countries, the East African single tourist visa will allow a visitor to enter any of the three countries that issued the visa and move freely within the other two.
The multiple-entry visa, which will be valid for 90 days, will cost about 75 euros.