"We agreed that in the next talks we will try to look for other alternatives."
Ahmad's statement came after the country's central government and the Sudan People's Liberation Movement (SPLM), the governing party in the south's semi-autonomous government, failed to agree over who is eligible to vote in the Abyei referendum.
Talks mediated by the international community were held last week in Addis Ababa, Ethiopia's capital, to discuss the issue but there was no progress.
The impasse over Abyei, which sits astride the oil-rich border between south and north Sudan, comes amid rising tensions between the two sides.
Omar al-Bashir, Sudan's president, warned recently that another civil war could break out if territorial disputes are not settled before a broader referendum on the south's possible secession, which is also set to take place on January 9.
Ahmad said Khartoum and the SPLM could reach "a conclusion on the final status of the Abyei area" without having to decide the matter through a referendum.
The referendum was promised by the Comprehensive Peace Agreement (CPA), which was signed in 2005, ending decades of the north-south civil war.
Under the agreement, the south formed its own government, which has limited autonomy and in which the north has a small representation. South Sudan is represented in the government of national unity, which is led by the Khartoum-based NCP.
Jalal Yousif al-Dagir, Sudan's minister of international co-operation, told Thursday's news conference the government would be open to a proposal to delay the referendum by four months or more.
"The referendum is not completely out of the window but it is apparent now that other solutions may be the real alternative," he said.
But Deng Arop Kuol, Abyei's administrator and a member of the SPLM, said the region's residents would not accept a delay and may hold their own vote without the central government's approval.
"A delayed vote is unacceptable," he said. "The people of Abyei are still holding out for the referendum to be held on January 9. If the government does not give them that option, we can have a self-run referendum."
Kuol declined to comment on whether he would accept any alternative to the Abyei vote, but said he and the SPLM would return to negotiations with the north in Addis Ababa, scheduled for the end of the month.
Al Jazeera's Haru Mutasa, reporting from the south's capital Juba, where Salva Kiir, the south's president, and leaders from numerous southern political parties, met to discuss the Abyei referendum.
"High on the agenda is the issue of Abyei. There have been closed-door meetings all day but the sentiment of people in the south seems to be that there is no way the referendum on Abyei is going to be postponed," she said.
"As far as they are concerned, it is meant to happen [on time] and they do not care what the Khartoum government is saying."
A delay of either the Abyei or secession referendum threatens to revive a new conflict between the two sides.
Both the north and south claim the oil-producing region and fought over it during the two-decades long war, in which around two million people died.
During a meeting with UN Security Council members in Juba last week, Susan Rice, the US ambassador to the UN, said Kiir has "warned that he fears the north may be preparing for war and may be moving troops southwards".
In a statement, Rice confirmed Kiir asked for a UN-run 16-km buffer zone along the north-south border.
Alain Le Roy, the UN peacekeeping chief, responded to the request on Friday by saying more UN troops would be moved to the regional border within weeks.
"We will increase our presence but only in some hotspots," Le Roy said. But according the UN mission in Sudan, UNMIS, the organisation did not have enough troops to create a buffer zone along the whole border.
UNMIS has also announced that it will step up checks on allegations made by the two sides that the other has launched a military buildup on the frontier.
Source: Al Jazeera & agencies