It said its "final offer" for the company was equal to 467p per share and would give GKN shareholders a 60 per cent stake in Melrose.
It said all recent attempts to engage in "constructive discussions" with GKN have been blocked.
The deadline for accepting the offer is 1pm on March 29.
GKN has a major operation in Telford and employs about 340 people at Hadley Park, making structural parts including wheels.
It has fought hard against the offer; offering to give back £2.5bn to shareholders and agreed to merge its car unit with US company Dana.
In a letter to GKN shareholders, Melrose chairman Christopher Miller criticised its takeover target's management team for refusing to hold discussions over the offer and slammed recent moves by GKN to sell its Driveline automotive business.
He said: "We have recently made attempts to engage in constructive discussion with the GKN board with a view to agreeing the terms of a recommended offer.
"Your board has unequivocally declined each time to enter into any such discussions with us and, as recently as March 9 was unwilling to even enquire on your behalf as to the terms of any revised proposal, a decision that no doubt you will find both surprising and disappointing.
"Instead of £1.4bn in cash and a majority investment in a stronger combined Melrose/GKN business, it appears the GKN board prefers a minority stake in a foreign business (Dana) with no GKN management involvement."
Mr Miller added: "In the interests of decisively resolving the future of GKN, we are pleased to set out the terms of an increased and final offer which would give GKN shareholders 60pc of the enlarged group and £1.4bn in cash."
The sweetener is necessary after GKN management revealed plans last week for a £4.4bn merger of its automotive unit with US peer Dana. This would leave GKN as a pure-play aerospace business, which historically trade at a multiple of almost 30 times, double automotive companies.
Melrose's previous offer was worth about 420p, below GKN's share price of 435.1p on Friday.