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."