Balfour Beatty, whose projects include a student accommodation block at the University of Edinburgh, said its executive chairman Steve Marshall met with his Carillion counterpart Philip Green a week ago to try and rescue the tie-up.
Mr Green agreed to alter the terms to give Balfour shareholders a dividend payment this year and find more cost-savings within the enlarged firm, on the condition that the group kept its US-based engineering business Parsons Brinckerhoff.
He also offered to cover the costs for businesses that have been bidding for Parsons, which Balfour put up for sale in May.
But Mr Marshall said yesterday: "It wasn't a credible proposal that was going to fix all the risks for Balfour Beatty shareholders."
He added: "For example, if bidders [for Parsons] were not prepared to carry on and if the merger then didn't go through, Balfour Beatty is basically left with a failed merger transaction and damage to Parsons Brinckerhoff."
Mr Marshall is leading the company while the hunt continues for a new chief executive. Andrew McNaughton left abruptly in May in the wake of the firm's third profit warning within 18 months.
The chairman said in relation to the CEO job: "There are plenty of top-line people who want to come along and talk."
Balfour said yesterday that the sale of Parsons Brinckerhoff is "well advanced", with third-round bids due soon.
The firm hopes a sale will allow it to return £200 million to shareholders and pay down its debts, and insists that there is "no strategic logic" to keeping it.
Carillion first approached its larger rival about a £3 billion merger several months ago, hoping to gain an international footing through Balfour's operations in 80 countries.
However, talks foundered in late July over the latter's plans to sell off Parsons. Balfour Beatty also argued yesterday that any deterioration in its revamped UK construction services business could undermine the synergies that Carillion expects to find within the enlarged company.
Balfour's pre-tax profits for the first six months of the year fell 53 per cent to £22 million, not including the cost of restructuring and impairments.
Revenues fell two per cent to £4.85bn, as underlying growth was wiped out by the rise of sterling against the dollar and the euro.
The company brought forward its results by two days following reports of renewed merger talks.
Balfour Beatty, which employs 260 people at Westpoint business park in Paisley, took steps to shake up its UK regional businesses last year, and said yesterday it expects all 17 of the remaining units to be profitable by January 2015 as order levels and margins improve.
"We simply need to keep the foot on the gas with all of the measures implemented," said Nicholas Pollard, the chief executive of UK construction services.
Debts at the firm were higher than executives had predicted at £364m, up from £173m, which Balfour said was caused by delays in getting the proceeds of its public-private partnership disposals.
Carillion noted Balfour's rejection yesterday but declined to comment further on its ambitions to merge.
The company is due to post its own results on August 20.
Shares in Balfour rose yesterday, closing up 5.7p at 242.9p, while Carillion's shares rose 4p to close at 324.7p.