The Hearts and Rangers kit supplier, which also supplies strips for the England and Northern Irish national teams and six English Premier League clubs, is set to become an affiliate brand of the world's biggest sportswear company.
The companies said Umbro, which has recommended the offer to its shareholders, is to fully retain its brand, and its headquarters will remain in the UK. Nike said the move would "significantly expand" its British presence.
Eunan McLaughlin, Nike's vice-president for Europe, Middle East and Africa, said the deal was "one of those merger opportunities made in heaven".
He said: "We want to invest in and grow the Umbro brand. We are in this for the long haul."
The deal comes less than a week after Umbro, based in Cheadle, near Manchester, said it had had a takeover approach.
On Friday, retailer JJB Sports bought a 10.1 per cent stake in the company in a move to protect its stake in the market for England football shirts.
Sports Direct International, owner of rival retailer Sports World, has a 15 per cent stake in Umbro.
Hearts Football Club, which signed a two-year agreement with Umbro in February, welcomed the news.
Campbell Ogilvie, operations director for Hearts, said: "It allows Hearts to benefit from the positioning of Umbro alongside Nike's huge international marketing and financial expertise and resources.
"The acquisition of the Umbro brand will go a long way to helping Nike become the world's No 1 football brand and this can only be good for Hearts."
Steve Makin, chief executive of Umbro, added: "This is an excellent deal for all our stakeholders: it provides great value for shareholders and exciting prospects for our colleagues, partners and customers around the world.
"We will be a stronger and better business as part of Nike and this deal will allow us to accelerate our existing growth strategy by leveraging Nike's global resources and expertise."
Nike, which only began its move into football in the early 1990s, has been steadily expanding its share of the market. The Umbro deal will see it become a closer rival for German-owned Adidas.
In addition to Umbro, Nike has a number of other affiliate brands, including Converse, which it bought in 2003, Nike Golf, Cole Haan, Exeter Brands and Nike Bauer Hockey.
Nike has assured the Football Association that Umbro's double-diamond logo will remain on the shirts of the England football team.
Nike said shareholders would receive 193.06p per share in Umbro - a premium of about 61 per cent to the closing share price of 120p on 17 October.