His comments come after his government accused the UN of hiding rebels and guns at their camps - which it denies.
More than 70,000 civilians are seeking shelter at UN bases across South Sudan after fighting broke out last month.
The UN says both government soldiers and rebels have committed atrocities in one of the world's poorest countries.
What started out as a political dispute between Mr Kiir and his former deputy Riek Machar on 15 December has escalated into full-scale conflict, with reports of ethnic killings.
Around 500,000 people have been displaced and the UN estimates that considerably more than 1,000 have been killed.
The UN is in the process of deploying an extra 5,500 peacekeepers to South Sudan, to bring its forces up to 12,500.
Read Mark's full report
Mr Kiir made his comments, broadcast on national TV, following the weekend capture of Bor from rebel forces.
The South Sudanese army also says it has recaptured the key town of Malakal after days of heavy fighting, though this is disputed by the rebels.
Friction with the UN followed an attempt on Sunday by the information minister to forcibly gain access to the UN base in Bor, where they believed armed rebels were hiding.
In a statement, UN chief Ban Ki-moon said he was "alarmed" and "disturbed" by the threat to staff of the UN Mission in South Sudan (Unmiss).
"We did not know that when the Unmiss was brought to South Sudan, they were brought as a parallel government with the government in South Sudan," Mr Kiir said.
"They fell short of naming the chief of the Unmiss as a co-president of the Republic of South Sudan.
"If that is the position of Ban Ki-moon, he should make it clear that he wants the UN to take over South Sudan."
Mr Kiir added that the UN should allow the government to search for guns among those seeking shelter in the camps - and hand over any guns, uniforms and government vehicles already in UN hands.
The Unmiss spokesperson told the BBC the UN was impartial and had a "zero tolerance policy towards weapons" in its camps.
"When people come to our sites with weapons, uniforms - military equipment of any sort - it has been given to us to make sure that the site have a civilian nature," Ariane Quentier told the BBC's Focus on Africa radio programme.
Ms Quentier said the UN was in discussion with the government about handing over such confiscated items and denied that the UN was sheltering rebels.
"It is very clear that we have a number of people that we cannot check individually but just the mere fact that we have 70-80% women and children is a clear indication that we're not sheltering rebels," she said, adding that she was sure relations with the government would improve.
"Like in every love story there are ups and downs - we may be a bit down these days but we are very confident that things will come back to normal."
Mr Kiir also accused other humanitarian organisations of supporting Mr Machar.
Talks to try to find a ceasefire are continuing in Ethiopia.
Mr Kiir reiterated his stance that he wants an unconditional end to hostilities - and will not release 11 political detainees, a key demand of  Mr Machar.
But he said the government had "space in its heart" to forgive Mr Machar.
He also urged his people to stop "the practice of a brother killing a brother" and "the practice of destruction".
Mr Machar has said he will not sign a peace deal unless Ugandan troops fighting alongside the government forces are withdrawn.
Nine Ugandan soldiers died in a rebel ambush at Gemeza a week ago, and 12 others had been killed in total since 23 December, Uganda's military spokesman, Col Paddy Ankunda, said on Twitter.
The Ugandan army was "not intent on staying for long in South Sudan", he said.
South Sudan President Salva Kiir has accused the UN peacekeeping mission of acting like a "parallel government" in his country.