Shares in GKN, one of the UK's biggest engineering firms, have soared after it rejected a bid that values the company at £7bn.
Melrose made the unsolicited offer of 405p a share on 8 January.
Shares in GKN, which makes wing tips for Airbus planes and parts for Porsche cars, jumped more than a quarter to 423p.
Its board said the cash and shares bid was "entirely opportunistic" and "fundamentally" undervalued GKN.
Shares in Melrose, a manufacturing turnaround firm, also rose sharply, adding 7% to 229p. Takeover rules mean it must make a firm offer by 9 February or walk away for six months.
Liberal Democrat leader Vince Cable called on Business Secretary Greg Clark to block the takeover: "GKN stands for long term investment in advanced manufacturing whereas Melrose are in the business of short-term financial engineering."
Last year, lower profit margins and cash generation prompted GKN to conduct a wide-ranging review of its business. The company also warned on profits after uncovering problems at its aerospace division.
On Friday, it said a new two-year strategy called Project Boost would significantly increase cash flow by cutting costs and expenditure along with tighter pricing control.
It also announced plans to split its aerospace and automotive divisions into separate companies, although the timing has not been confirmed.
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.
"Historically, the pension deficit has held the group together, but with the sprawling footprint likely to have contributed to recent profit warnings, the reasons for divorce now seem to outweigh the costs of splitting," he said.
"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."
GKN said on Friday that Anne Stevens, its interim chief executive, would remain in the role permanently.
Kevin Cummings had originally been due to take up the role on 1 January, but in November the company announced it was looking for "alternative leadership".
The decision followed a £130m writedown in GKN's aerospace division, which Mr Cummings had been running.
"Operational challenges" at its North America-based aerospace business were one factor behind a profit warning in October.
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.
The company's origins can be traced back to the Dowlais Iron Co, near Merthyr Tydfil in South Wales, in 1759. The town's ironworks supplied rails to railways in the UK and abroad, making it the world's largest by 1845.
The Patent Nut & Bolt Company, founded in Birmingham in 1856, was combined with Dowlais in 1900 in a new group called Guest, Keen & Co.
Two years later, it took over Nettlefolds Ltd to become Guest, Keen & Nettlefolds.
GKN was heavily involved in wartime production in the first half of the 20th Century, during which time the company made its first move into the emerging motor industry.
Melrose specialises in buying up under performing companies.
The firm first listed in 2003 with a value of £13m, but in the 15 years since then its value has rocketed. Today the FTSE 250 company is valued at almost £4.4bn.
Its acquisitions have included Dynacast, which it bought from private equity firm Cinven in 2005 and sold six years later for four times its original investment.
In 2012 Melrose bought utility metering firm Elster for £1.8bn and sold it three years later to Honeywell for £3.3bn.
Last August the firm acquired US company Nortek for £2,2bn.