The offer is likely to be rejected by Raila Odinga, leader of the opposition Orange Democratic Movement (ODM), because the post is likely to be non-executive.
Peter Anyang Nyong’o, secretary-general of the ODM, said that he expected an agreement by the end of Friday.
"The negotiators are at the table, so we must give them time," he said.
"If you are going to have real power-sharing, you must separate the head of state from the head of government ... Which has been a proposal made by Kenyans since 1992.
"It is a response to the demands and wishes of Kenyans for more effective and democratic government so that the state runs properly."
Kilonzo, negotiating at talks led by Kofi Annan, the former UN secretary general, added: "What we are discussing now is the post's functions, responsibilities, nature of appointment and so on.
A senior opposition official, speaking on condition of anonymity, said there was discussion of two roles for the prime minister.
"One of them is some sort of chief minister ... or there is a head of government who is answerable to parliament and who is appointed and his conduct is regulated by the national assembly. You have a real separation of powers," the official said.
He said the interim post would come with a sunset clause that causes the position to expire either when parliament is dissolved or when a new constitution is enacted.
John Cookson, Al Jazeera's correspondent in Nairobi, said that it could be a significant move forward, as the position had been abolished many years ago and reinstating it is a key demand of the ODM.
The ODM has demanded the post and a 50-50 split in cabinet positions.
Cookson said getting the required constitutional change in order to implement the post through parliament could, however, lead to months of prevarication.
Meanwhile in Brussels, the International Crisis Group (ICG), a peace advocacy body, said that while dialogue was taking place in order to solve Kenya's political tensions, fresh violence was fomenting.
"Calm has partly returned, but the situation remains highly volatile... Armed groups are still mobilising on both sides," the report said.
"Extremists and militia are preparing for new confrontation on both camps and ODM believes that if international mediation fails, its only protection against repression and hope for a settlement will be its capacity to raise the stakes through violence," the report said.
The ICG report said the banned Mungiki sect, whose membership consists of mainly Kibaki's Kikuyu ethnic group, jumped into action moments after the president was re-elected on December 30, beheading and mutilating members of rival communities in the capital's slums.
"The uneasy calm in Kenya should not be misunderstood as a return to normalcy," ICG said.
"The protracted political crisis has deep roots and could easily lead to renewed extreme violence."Athletic elite
|ICG said the violence has shattered Kenya's|
image of stability in the region [AFP]
The report also identified wealthy Kenyan athletes as a new tier of authority, emerging as benefactors of tribal groups, taking over the role from traditional elders.
ICG said that athletes, most of whom have a military background, were reportedly training and sometimes commanding the gangs to fight.
Francois Grignon, director of the ICG's Africa programme, said in an article with The Times: "We are very confident that they are part of the equation.
"They are a rising elite who see their leadership potential and economic prospects being curtailed by the fact that they are politically marginalised nationally."
Athletics officials dismissed the allegations as rumours designed to further enflame tribal tensions.
"The violence has shattered Kenya's reputation for stability," the ICG said.
"The grisly images ... illustrated the fragility of a national fabric in which the disparity between rich and poor remains one of the world's biggest."