Bashir's offer was delivered to Kiir by a Sudanese delegation that visited Juba on Tuesday, and comes just days after the UN Security Council adopted a resolution that gave South Sudanese warring sides one month to reach a peace deal or face possible sanctions.
The Sudanese delegation was led by Foreign Minister Al-Dierdiry al-Dhikheri and included Salah Ghosh, the chief of Sudan's powerful National Intelligence and Security Services (NISS).
"The delegation delivered a verbal message from President Omar al-Bashir to President Salwa Kiir regarding an initiative to support efforts to achieve peace between the conflicting parties in South Sudan," Sudanese foreign ministry spokesman Garib Allah Khadr said in a statement.
Bashir offered to "host a meeting between Kiir and South Sudanese opposition leader Riek Machar," he said.
"President Kiir welcomed the initiative and affirmed his government's readiness to participate in the meeting."
Khadr did not say when a meeting was scheduled or whether Machar had agreed to it.
Bashir's offer comes even as the regional Intergovernmental Authority on Development (IGAD) group is mediating peace talks between Kiir and Machar.
South Sudan, the world's youngest nation formed after splitting from the north in 2011, descended into war in December 2013 when Kiir accused his former deputy Machar of plotting a coup.
A peace deal was signed two years later but collapsed in July 2016 when fresh fighting in the capital Juba forced Machar into exile. The renewed violence spread across the country, and numerous new armed opposition groups have formed, further complicating peace efforts.
Efforts to revitalize a 2015 peace agreement resulted in a ceasefire in December which lasted just hours before warring parties accused each other of breaking it.
