Adidas-Salomon AG, the world’s second-largest sporting-goods company, agreed to buy Reebok International Limited for about $3.8 billion to narrow the gap with Nike Incorporated.

Adidas is offering $59 for each share of Canton, Mass.-based Reebok, 34% more than Monday’s closing price. The combined company would have about $11.1 billion in sneaker and sports equipment sales compared with Nike’s $13.7 billion. Adidas’s stock rose to the highest in more than seven years, while Reebok’s climbed 30% to a record.

The purchase is the largest under Chief Executive Herbert Hainer and will double Adidas’s sales in America, the world’s biggest market for sports gear. Herzogenaurach, Germany-based Adidas, whose shoes are worn by soccer star David Beckham, will have about 20% of the American market, still behind the 40% held by Nike, whose endorsers include golfer Tiger Woods.

“The U.S. market is historically where they’ve been under-represented, and it’s been difficult to gain share,” Mark Hargraves, who helps manage $7.1 billion including Adidas shares at Framlington Group PLC in London, said. “It propels them up.”

Adidas shares rose 7.4% to $195.27 in Frankfurt, the highest since July 1, 1998. Shares of Reebok, whose endorsers include Houston Rockets basketball player Yao Ming and rapper 50 Cent, were up $13.19 to $57.14 at 4 p.m., the highest since the company’s 1985 initial public offering.

Adidas also said today that second quarter profit rose 34% to $115.8 million from $86.3 million a year earlier.

Analysts were expecting $101 million, according to the median of five estimates gathered by Bloomberg. Sales gained 8% to $1.87 billion.

Reebok Chief Executive Paul Fireman, who built a small British shoemaker into a company with $3.78 billion in yearly sales, stands to make about $437.8 million from the deal. The CEO owned about 7.42 million Reebok shares as of June, and his wife, Phyllis Fireman, had about 6.02 million shares as of February, according to Securities and Exchange Commission filings.

Phyllis may make $355.2 million from the planned sale.

Mr. Hainer said he doesn’t expect a “significant work-force reduction” after the acquisition as he intends to have the two businesses run separately. Mr. Fireman, who founded the company in the 1970s, will continue to run the brand.

The acquisition will boost Adidas’s earnings in the first year and save $123.2 million to $154 million the first three, Mr. Hainer, 51, said. Adidas’s profit margin has averaged about 2.7% a year the past five years, while Nike’s is about 6.4%.