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NEW YORK — Sprint Nextel has increased its bid for full control of its wireless-network partner Clearwire, seeking to persuade shareholders of the Bellevue company to reject a competing proposal from Dish Network.

Sprint offered $3.40 a share, up 14 percent from a previous bid of $2.97 a share, the company said Tuesday in a statement. That compares with Dish’s proposal of $3.30 a share. Clearwire, which was scheduled to vote on Sprint’s earlier offer Tuesday morning, delayed the decision until May 31.

The sweetened bid follows a showdown with Clearwire investors, who balked at the original terms. Even though Sprint already owns slightly more than 50 percent of the business, the full takeover would give it control over Clearwire’s valuable spectrum — something it needs to bolster its own network.

Sprint realized it had no choice but to boost its price, said Roger Entner, an analyst at Recon Analytics. “It was a game of chicken, and they blinked,” Entner said. “They need to have Clearwire integrated — there’s no way around it.”

Clearwire shares climbed 4.3 percent to $3.40 Tuesday after Sprint raised its offer. The shares had already been trading well above the earlier $2.97 bid, a sign investors saw the increase coming.

Bob Toevs, a Dish spokesman, declined to comment on whether the satellite carrier would raise its bid for Clearwire.

Sprint, the third-largest U.S. wireless carrier, needs support from the majority of Clearwire’s Class A shareholders to gain control of the remaining 49 percent of Clearwire it doesn’t own. The move would wind down an ambitious joint venture with Clearwire that had attempted to build a nationwide wireless Internet network.

Begun in 2008, the project was backed by $3.2 billion in investments from Google, Intel and cable-TV companies. After losses piled up, partners such as Google and Time Warner Cable sold their stakes for a fraction of their original value.

While Overland Park, Kan.-based Sprint has the upper hand because it already owns more than half of Clearwire, it has faced opposition from investors such as Mount Kellett Capital Management and Crest Financial.

“With a sweetened deal they should win the votes they need,” Entner said. “There are some religious opponents they won’t be able to convert, but the $3.40 should get them an approval.”

Crest said Tuesday in a statement that Sprint’s new offer is still not enough. The Houston-based investment firm said it sent a letter urging Clearwire’s board to reject the new bid, arguing it should have a separate, competitive process to seek proposals.

Clearwire has said it faces a cash crunch and needs at least $1.7 billion to keep operating. Shareholder-advisory groups Institutional Shareholder Services and Egan-Jones Ratings both endorsed Sprint’s earlier bid, citing Clearwire’s dim prospects as an independent company.

Glass, Lewis & Co., another proxy-advisory firm, disagreed, saying Sprint hadn’t made a compelling case that its offer was the best option. Clearwire investors such as Crest argued that the company’s assets were being undervalued. Crest even offered to lend the company money itself to help keep it afloat.

Sprint made its bid for Clearwire after agreeing to a deal with SoftBank in October. In that transaction, Tokyo-based SoftBank would acquire 70 percent of Sprint for $20.1 billion. The takeover would help SoftBank expand into the U.S. and give Sprint an $8 billion cash infusion.

Dish, the Englewood, Colo.-based satellite provider controlled by billionaire Charlie Ergen, is making a separate attempt to thwart the SoftBank acquisition. He bid $25.5 billion for Sprint last month, part of a strategy to expand into the mobile-phone business.

Sprint said Tuesday that it will give private financial data to Dish, letting the company build a case for its takeover offer.

Sprint said it got a waiver from Softbank on Monday to provide the nonpublic information to Dish.

Sprint said its board hasn’t made a decision on Dish’s bid and continues to back SoftBank’s proposal, though it has the right to terminate the agreement for a superior offer.