Dish and Sprint have been fighting publicly over Clearwire since January. Clearwire has vast troves of valuable wireless airwaves that both companies want to use to help them compete in the wireless-services market.
Sprint said it now has support from shareholders with 45 percent of Clearwire's minority shares, just shy of the more than 50 percent it needs to take over the company.
For example, Sprint said it has commitments from a group of activist shareholders that own about 9 percent of Clearwire's voting shares to support the deal.
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These shareholders—Mount Kellett Capital Management, Glenview Capital Management, Chesapeake Partners Management and Highside Capital Management—had teamed up to negotiate together for a higher price.
In addition, Clearwire promised to pay a break-up fee of $115 million if the current agreement was terminated.
Sprint raised its bid just a few days before shareholders in the smaller wireless company were due to vote on its previous $3.40-per-share offer.
Shareholders had complained that Sprint's previous offer was too low even before Dish made its counterbid. Analysts and investors had said that Sprint would need to raise its bid or risk a contentious relationship with Dish as a minority shareholder.
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Dish was not immediately available for comment.
The news came two days after Dish Chairman Charlie Ergen decided to back out, at least for now, from a battle with Japan's SoftBank to buy Sprint itself.
Clearwire shares closed up 34 cents, or 7 percent, at $5.04 on Nasdaq. Sprint shares rose 7 cents, or 1 percent, to $7.07 on the New York Stock Exchange.