Melrose confirmed this morning it has increased its offer for FTSE 100 giant GKN to £8.1bn.
Melrose responded by taking its offer directly to GKN investors, ramping up the pressure on GKN's newly appointed chief executive, Anne Stevens.
GKN shares rose almost 2% to 442p in early trading on Monday.
That reflects considerable uncertainty on whether the deal will happen.
Melrose, a UK-based turnaround specialist, said on Monday that GKN investors would receive 81 pence in cash for every GKN share they hold, plus 1.69 new Melrose shares under its offer, upping the equity part of the deal from 1.49 new Melrose shares.
On Friday, GKN struck a £4.4bn deal to merge its automotive business with the USA firm Dana in an attempt to fend off Melrose.
The board of GKN, a maker of parts for use by vehicle and plane makers, has been battling to show shareholders that it can create more value for them with its own plan for the company since January when Melrose first approached it.
"Your board has unequivocally declined each time to enter into any such discussions with us and, as recently as Friday 9 March, 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".
Melrose said that Monday's offer, up from a previous one of £7.4bn, was its best and final offer and would not be improved "under any circumstances".
Mr Miller said: "We are nearing the end of the customary offer timetable and it is now time for you to decide".