The listed, Wolverhampton-based group said it would offer Balfour's shareholders a 36 per cent premium for their shares.
In a statement to the stock exchange, Carillion said it hoped that Balfour would re-engage in merger discussions after calling a halt to the talks at the end of July and then rejecting a later, revised proposal from the Black Country firm.
Carillion, which built the Library of Birmingham, said it would offer a 58.268 per cent share for Balfour Beatty shareholders based on the current undiluted ordinary share capital of each of Balfour Beatty and Carillion.
It would also offer a cash dividend or equivalent of 8.5 pence per Balfour Beatty share, totalling £59 million, which would value Balfour Beatty at £2.08 billion.
Balfour Beatty, which is building the extension to the Midland Metro in Birmingham city centre, had previously agreed to a 56.5 per cent share for Balfour Beatty shareholders.
Carillion said this improved proposal represented a premium of 36 per cent to the 'one-month volume weighted average price' prior to July 24 when the companies made a joint announcement about the fact they had entered talks.
Balfour Beatty announced in May that it wanted to sell its consultancy arm Parsons Brinckerhoff but Carillion has been steadfast in its belief that a merged group should have Parsons within it.
In response to this latest development, Balfour Beatty said: "The board notes the announcement made by Carillion.....and will make a further announcement in due course.
"There can be no certainty that an offer will be made by Carillion for Balfour Beatty nor as to the terms of any such offer."
The current deadline for Carillion to announce a firm offer, or withdraw, is 5pm on August 21 but it is hopeful of gaining an extension to this.