Not quite two years after hiring new leaders and setting a new strategy, the board of Digi International Inc. is suddenly facing a choice between staying on that path or selling out to a bigger rival.

Over the weekend, the board for a second time in a week rejected a takeover offer by Belden Inc., which makes telecom equipment as does Digi. In a statement issued Sunday, Minnetonka-based Digi said Belden’s offer of a 20 percent premium to its share price was too low.

Investors got their first chance to react to the back-and-forth between the firms Monday, and shares in both companies rose. Neither company had made their discussions public until after the stock market closed Friday.

Digi shares closed up 15 percent at $13.40 a share, the highest level since July 2011. Belden shares rose 2.2 percent to $71.22.

Belden, based in St. Louis, first approached the Digi board on Nov. 4 with a cash offer of $13.82 a share, or about $380 million for the entire company. Digi rejected that offer last Tuesday. After Belden made the offer public late Friday, the Digi board again conferred and rebuffed the advance, following up with Sunday’s statement.

“The Digi board and management are confident in Digi’s stand-alone prospects and believe that the company is solidly positioned to create significant value for Digi shareholders,” the statement said.

Digi, which has about $200 million in annual revenue, has seen its sales and profits grow since the arrival in late 2014 of chief executive Ron Konezny. He accelerated the company’s drive into wireless modules that can be used to hook up almost anything to the internet, pushing into markets known in the tech industry as machine-to-machine communication and the “internet of things.”

He also started shifting Digi away from customized products, streamlining its portfolio to simplify manufacturing and inventory management. Earlier this year, he hired a new product chief to strip down the portfolio even more, and more recently appointed a new top sales executive.

In the statement issued Sunday, Digi said it “has been successfully executing on its strategy to simplify and scale its strong core business while focusing on three key priorities to drive growth.” It identified the three priorities as consistent profitability, steady sales growth and focusing on hardware products.

A call to a Digi executive on Monday wasn’t returned. Belden made no public response on Monday to Digi’s rejection.

“The offer provides Digi’s shareholders certainty of value and immediate liquidity and removes financing, market and execution risks,” John Stroup, Belden’s chief executive, said in a telephone conference with analysts Friday night.