Aberdeen-headquartered Wood Group is mulling over a final offer worth almost £1.6bn from Dubai-based rival Sidara.
The fourth offer, of 230p per share, is up from previous offers of 220p per share, 212p per share and 205p per share.
Wood's board had previously said offers from Sidara "significantly undervalue the group and its prospects".
Following the latest offer, the firm said: "The board, together with its financial advisers, is evaluating the final proposal and a further announcement will be made in due course."
Wood also confirmed no engagement had been with Sidara since the company made its first approach.
Currently, under the Takeover Code, Sidara has until June 5 to either announce a firm intention to make an offer or to announce it has no intention to make an offer.
However, Wood reports that the Middle Eastern company "does not believe that its proposal can be progressed unless the board of Wood engages with Sidara and an extension to the deadline is granted".
The latest approach is 52% greater than Wood's average share price over the last year, before Sidara's interest became known.
Since news broke of Sidara's final bid, shares in Wood fell by 2.5%.
Who is Sidara?
The Lebanese-founded design and engineering business rebranded to its new name last year.
It's reported that it sold down a $1.4bn (£1.12bn) stake in Australian engineering services group Worley in April, to prepare to approach Wood.
The company, founded in 1956, generates about half of its $2.3bn (£1.84bn) revenue in the US and was believed to have been interested in Wood as a channel to grow that further through new oil and gas projects and renewable energy.
Around a quarter of Wood's revenue last year was accounted for in the US.