Kan, heavily criticized during Japan's recovery from this year's earthquake, tsunami and nuclear disaster, survived a no-confidence vote in Parliament but says he will quit in August.
The Yomiuri Shimbun reported Friday that some figures in the Democratic Party of Japan, including acting party president Yoshito Sengoku, are backing Noda.
But other DJP members from the factions led by former Prime Minister Yukio Hatoyama and Ichiro Ozawa are pushing Agriculture Minister Michihiko Kano, sources said.
Former Environment Minister Sakihito Ozawa and Shinji Tarutoko, chairman of a Parliament committee, may also seek the leadership.
Noda is seen as representing continuity.
Issues he is facing as finance minister, including an emergency reconstruction budget and reform of the pension and tax systems, will be central in a new administration.
Noda would not comment to reporters Thursday.