The company, which has been pursuing a £3 billion tie-up with Balfour Beatty since July, yesterday offered to give Balfour shareholders 58.3 per cent of the combined company, up from its previous proposal to hand them 56.5 per cent.
Balfour, which employs 260 people in civil engineering and construction at its site in Paisley, has already rejected two merger packages from its smaller rival. The FTSE 250-listed firm acknowledged yesterday's proposal and said it would respond in due course.
Carillion has until 5pm tomorrow to make a formal bid or walk away for at least six months under Takeover Panel rules, unless the companies ask the regulator for an extension.
A sticking point between the two firms is Parsons Brinckerhoff, the US consultancy business that Balfour put up for sale in May. Carillion has refused to back down over its desire to keep the business, but has offered to cover costs for companies that have already bid.
Balfour, which lost its chief executive Andrew McNaughton in May after a string of profit warnings, has insisted that there is no logical reason to keep Parsons and said it planned to hand shareholders £200 million once it was sold.
Carillion, meanwhile, has agreed to pay Balfour's investors £59m in dividend payments if the merger goes ahead.
The two sides have also squabbled over the amount of cost savings that could be made by combining the companies.
David Cumming at Standard Life, which owns shares in both firms, on Monday said Balfour should reopen talks with Carillion if the latter were to "modestly improve its terms".
Carillion said it has tabled its latest plan following talks with Balfour's major shareholders.
Shares in Balfour Beatty closed 3.2 per cent higher yesterday at 256p, the highest price since its latest profit warning in May.