The takeover specialist has raised its cash-and-share offer for the United Kingdom engineer by just over a tenth to 7.9 bln pounds.
Earlier, Melrose set out the terms of its increased bid - at 467p per share - which would give GKN investors a 60% stake in the expanded Melrose. Shareholders also would be entitled to receive the 6.20p final GKN dividend.
The deadline for accepting the offer is 1pm on Thursday, March 29. Under the prior deal Melrose offered 1.49 Melrose shares in addition to the 81 pence in cash and dividend entitlement.
Earlier Monday, GKN had said it "evaluating" the revised Melrose offer.
As of 08:11 GMT, GKN's share price had added 1.75 percent to 442.70p, outperforming the benchmark FTSE 100 index which now stands 0.18 percent higher at 7,237.72 points.
"The GKN Board is suggesting this is the best option for shareholders, employees, pensioners and other stakeholders".
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By 1300 GMT on Friday, which was the first closing date of its offer, Melrose had received acceptances from investors holding only 5.76 percent of GKN stock, the turnaround specialist said. Further, at Melrose's midday Monday price of 215.70p, the revised offer was only worth 445.50p.
According to the FTSE 100 firm, the new bid does not reflect the value of GKN's "world class aerospace business" or the benefits of combining GKN Driveline and Dana Incorporated - GKN last week agreed to merge its auto parts business with the USA firm in a deal worth $6.1bn (£4.4bn).
GKN said its board unanimously rejected the offer, once again asserting that it "fundamentally undervalues" the business and its prospects.
GKN shares fell 2.2 percent to 425.5 pence by 1556 GMT, compared to the 467 pence per share initial value of Melrose's final offer.
"Melrose is not the right owner of GKN", Chairman Michael Turner said.
"Instead of £1.4 billion 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". Winning new business in our markets would be more hard if customers were uncertain as to the identity of their future long term partners.