That brings Softbank's total bid to $21.6 billion for the nation's third-largest carrier, which is still short of the $25.5 billion offered by the rival Dish Network in April.
Sprint, based in Overland Park, Kan., said already said it's determined that Dish will not be able to come up with an offer superior to Softbank's. It ended discussions with Dish and gave it a June 18 deadline for a best and final offer.
The revised deal announced late Monday by Softbank suggests that it still seeks to mollify any Sprint shareholders who are not yet sold on the deal.
The new offer pushes even more cash to Sprint shareholders: $16.6 billion, up from $12.1 billion. In exchange, Softbank will own about 78 percent of Sprint, compared with a previous 70 percent.
Shares of Sprint climbed 3 percent, or 22 cents, to $7.40 Tuesday before markets opened.
Sprint's second largest shareholder, Paulson & Co., said it will vote all its shares in favor of Softbank's sweetened offer. And Softbank, even before the revised offer this week, had secured the endorsement of shareholder advisory firm Institutional Shareholder Services.
ISS believes the Softbank bid would ease Sprint's debt burden and provide enough cash to improve its network. ISS noted that mobile data in Japan travels nearly twice as fast as mobile data in the United States, and Softbank's expertise could eventually make Sprint's network faster than AT&T and Verizon.
Sprint Nextel Corp., with more than 55 million subscribers, trails both Verizon Wireless and AT&T.
Softbank Corp. is a holding company with investments in Internet and telecom businesses. It was the first carrier to offer the iPhone in Japan.
The Treasury Department's Committee on Foreign Investment has already cleared Softbank's bid, saying it hadn't found any national security reasons to prevent it. There were concerns that Softbank's use of Chinese networking equipment could open up U.S. networks to snooping by China.
The Softbank deal also needs approval from the Federal Communications Commission, but Sprint expects the transaction to close in early July.
Last month, Dish Chairman Charlie Ergen told investors and reporters that, given the benefits of owning Sprint, Softbank should be paying more for the company.
Softbank's CEO Masayoshi Son fired back, saying that Dish's $25.5 billion offer was based on "incomplete and illusory" numbers.
Sprint shareholders had been scheduled to vote Wednesday on the previous Softbank offer, but the companies pushed that back to June 25 after Softbank raised its bid.