Umbro's board of directors unanimously recommended that shareholders accept the offer, which values the company at $582m (£285m) or 195 pence per share.
Umbro supplies the kit for the England football team and six other Premier League clubs including Birmingham, Blackburn, Everton, Sunderland, West Ham and Wigan.
Umbro chief executive Steve Makin said the acquisition would allow the business to accelerate its "existing growth strategy by leveraging Nike's global resources and expertise". He added: "We look forward to taking Umbro to new levels with Nike's support."
The acquisition will help Nike to significantly grow its football revenues, a key growth category for the company. Since the early 1990s, revenue has grown from about $40m to $1.5 billion.
Nike president and chief executive Mark Parker said: "Umbro is a brand with a powerful heritage and deep experience in the world's most popular sport and biggest football market. The dynamic alignment of Umbro and Nike, with our complementary strengths and numerous ways to segment and grow the market, will lead the game at every level throughout the world."