Shares in Melrose Industries were 10.71% higher at 238.1p in early trade on the London Stock Exchange.
The company said it intends to separate the businesses.
Under the stock market rules, Melrose is now required to either announce a firm offer by 5pm on 9 February, or announce that it is no longer interested.
Last year, lower profit margins and cash generation prompted GKN to conduct a wide-ranging review of its business.
GKN also announced interim chief executive Anne Stevens would become the new boss of the company with immediate effect after its incoming chief executive left the group in November before taking up his new role.
Its board, in tandem with financial advisers Gleacher Shacklock, JP Morgan Cazenove and UBS, "unanimously rejected" the proposal, which it deemed "entirely opportunistic" and to "fundamentally undervalue" the company and its prospects.
GKN said each business would have distinct investment profiles and capital allocation policies.
Nicholas Hyett, an analyst at Hargreaves Lansdown, said the split had been "on the cards for years" because there was little crossover between the two businesses.
"The money to be made from a split is likely to have been what drew turnaround specialist Melrose to the table in the first place - the challenge for newly confirmed chief executive Anne Stevens is to deliver a better result for shareholders than the 405p she turned down today".
The decision followed a £130m writedown in GKN's aerospace division, which Mr Cummings had been running. That write-off, associated with GKN's U.S. aerospace business, will be "nearer the upper end" of an £80-130 million range, GKN indicates.
The aerospace division was bolstered in 2009 when GKN bought the Airbus wing manufacturing and assembly plant at Filton, near Bristol, and Fokker Technologies in 2015.
A bid battle is now likely as Melrose, which specialises in buying struggling businesses, turning them round and selling them at a profit with the proceeds given back to shareholders, will pursue its interest despite GKN's opposition. The town's ironworks supplied rails to railways in the United Kingdom and overseas, making it the world's largest by 1845.
"If Melrose's bid is successful it will blow a hole in the government's hope of developing a coherent industrial strategy and signal that government ministers are happy to see yet another British company fall prey to vulture capital".
Speculation has been ongoing for a while as to whether the firm would be a takeover target or decide to break itself up.
Last August the firm acquired USA company Nortek for £2,2bn.