Ministers had insisted the anticipated budget of £235m-£295m would not be breached, but sources close to the paper have now reported the amount will total £350m
It is also thought that the new route, which will run between Edinburgh and Tweedbank, will open a year later than planned in 2015.
The Scottish Government’s Transport Scotland agency has announced a contract with Network Rail to build the line would be signed by Wednesday.
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The original plan for the project included a private sector consortium to fund the railway, after two of the three short-listed bidders pulled out.
The project is already delayed, as the main construction work was supposed to have started by autumn 2011.
The estimated costs have also been exceeded. £54m has been spent on other work, including nearly half on land purchase, while £10m has gone on advanced construction and site investigations.
As of yet, no start date has been announced.
Scotland on Sunday believe the tender could be awarded to Dutch firm BAM Nuttall.
One source told the paper: "The general view in the industry is that both the programme and cost were very ambitious, and will have changed significantly.
"Increases in inflation and the cost of materials could well mean the total increasing to around £350m."
(IT)
