Prime Minister Shinzo Abe will make a final decision soon, in consultation with Finance Minister Taro Aso, and submit the government's nominee for parliamentary approval next week, the paper said on Saturday.
Kuroda, Japan's former top financial diplomat, has been regarded as among the key candidates to succeed incumbent BOJ Governor Masaaki Shirakawa, who leaves the central bank with his two deputies on March 19 after a five-year term.
Former top financial bureaucrat Toshiro Muto and Kazumasa Iwata, a former government economist and deputy BOJ governor, have also been floated as strong candidates to be the next BOJ head.
Kuroda is now leading the race, Asahi cited Finance Ministry officials and senior government officials as saying.
Abe, now in Washington, said he planned to seek the backing of a junior coalition partner and opposition parties over the nomination of a new BOJ governor on his return from his U.S. trip on Sunday.
The nomination of the next BOJ governor must be approved by both houses of parliament. Abe's ruling Liberal Democratic Party lacks a majority in the upper house and needs support from opposition parties to get the nomination passed.
As Japan's top financial diplomat from 1999 to 2003, Kuroda aggressively intervened in the exchange-rate market to weaken the yen to support the country's export-reliant economy.
He has also been a vocal advocate of more aggressive monetary easing, a view in line with that of Abe, making him a strong candidate to head the BOJ.
If Kuroda were to be chosen as next BOJ governor, he would be cutting short his term as head of the ADB, which could weaken Japan's standing as the country that traditionally provides the head of the organization established in 1966.
(Reporting by Leika Kihara; Editing by Jon Hemming and Peter Cooney)