At one point, shares in the company were up 35%, but they then fell back to close 13.7% higher.
Without citing sources, Kyodo News said a state-backed fund may invest in Sharp if Japanese lenders agreed to write off an unspecified amount of its debt.
Kyodo said the banks were expected to consider the plan, among other options.
In May, the firm agreed to a 200bn yen ($1.7bn; Â£1bn) bailout from banks and announced more job cuts.
Mizuho Bank and Bank of Tokyo-Mitsubishi UFJ each provided a 100bn yen investment in Sharp. It is also received 25bn yen from Japan Industrial Solutions to help fund its "business growth strategy".
The bailout was the firm's second in three years and was announced as Sharp reported larger-than-expected losses for the year to March 2015.
The report said that if Sharp's debts were cut, the government-backed Innovation Network Corporation of Japan -  which was created to help increase the competitiveness of Japanese firms - might spin off the firm's troubled liquid display business (LCD).
The firm's LCD unit saw a loss of 12.7bn yen ($102m; Â£67m) for the quarter to September. In October, Sharp cut its earnings outlook for the full year to March 2016 to 10bn yen from 80bn yen.
The tech giant, which makes televisions, solar panels, tablets and phones, has been facing stiff competition from cheaper Taiwanese, South Korean and Chinese electronics firms.
Last week, Sharp said it wanted its workers to buy its products, in a attempt to boost sales.
The electronics giant said it was not mandatory for staff to buy its goods, but confirmed it wanted workers to choose its products over that of its competitors.
Sharp has also announced cost-cutting plans such as selling the company's Osaka headquarters and pulling out of the television business in North America.
Shares in troubled electronics maker Sharp have surged on a report that its lenders might waive some of its debts to allow new investment.