A statement from Ugandan President Yoweri Museveni's office said Kiir told the Ugandan leader that both the government in Khartoum and his Sudan People's Liberation Movement/Army (SPLM/A) "are willing to have joint operations with the UPDF (the Ugandan army) against rebels remnants".
Museveni's office said Kiir's pledge followed a proposition by the Ugandan leader for a joint operation involving the SPLM/A, the Sudanese army and the Ugandan military as a way to completely wipe out the LRA and its leader, Joseph Kony.
"Kony has been weakened and has fled to southern Sudan with only 80 people. [The] pressure should be kept on so that he does not get time to regain strength," it said.
Khartoum and Kampala signed a deal in 2002 that allowed the Ugandan army to conduct raids in remote southern parts of Sudan following repeated complaints that Khartoum was harbouring the notoriously brutal LRA.
The Ugandan army made some gains by destroying some of the rebels' bases and capturing weapons, but the insurgents fled towards the border and re-entered Uganda, carrying out a spate of brutal attacks and killing civilians in the northern region.
Last month, Kiir issued a stern warning to the LRA rebels to leave their hideout in southern Sudan's jungles, and urged Kampala to continue to pursue the stalled peace process.
Kiir, who visited Uganda on Wednesday, also discussed with Museveni issues relating to transport between the two countries, including a proposal to link the two countries with a railway line.
"President Museveni suggested that the best way is to extend the railway line from Packwach through Arua [all in northwestern Uganda] to Yei and Juba [Southern Sudan]," the statement said.
It added that the two also discussed the probe into the crash that killed former Sudanese First Vice-President John Garang in July, and said Kiir had expressed satisfaction with the ongoing probe into the helicopter crash.
An international team is currently probing factors that led to the 30 July crash, in which the Ugandan presidential helicopter carrying Garang came down near the Uganda-Sudan border, killing everyone on board.
Museveni, who supported the SPLM/A during its 21-year struggle against successive Islamic governments in Khartoum, urged Kiir to stick to the Comprehensive Peace Agreement that ended Africa's longest-running civil war.
"The president urged the SPLA to continue with the peace pact that was signed between them and the Sudanese government," Onapito Ekomoloit, Ugandan presidential spokesman, said.