Glasgow-based Weir went back to Helsinki-listed engineering firm Metso on May 20 with an improved proposal on its earlier approach in March.
It said the all-share deal represented a 13% hike on the previous proposals. It valued Metso's shares at 30.49 euros (£24.77) each, making the company worth about £3.7 billion.
However, Weir - whose market value is about £5.5 billion - said the Metso board "did not engage" on this second proposal and that the Finnish company rejected it as significantly undervaluing the business.
The group said: "Weir believes it made a compelling proposal but remains financially disciplined and therefore does not intend to pursue this opportunity further at this time."
FTSE 100-listed Weir has more than 15,000 employees across 70 countries. Shares slipped by nearly 2% on today's announcement.
The firm, which is a supplier of pumps and valves to the oil and gas industry, offered 0.95 Weir shares for each Metso share. It said the offer represented a 34% premium to Metso's undisturbed share price prior to its initial proposal on March 26.
Weir, whose clients include energy giant EDF and oilfield services group Schlumberger, added that the combined group would have also been able to make savings of at least £150 million a year once the merger had been completed.
Metso, which employs 16,000 staff in 50 countries, provides engineering services to the mining, oil and gas and construction industries.