According to STV, the final cost of the project is said to be almost £30m.
The City of Edinburgh was in disagreement with contractors Lagan Construction Limited over time delays and contract "interpretation" on the Water of Leith Flood Prevention Scheme.
A report, which is expected to be presented at a council meeting later this week, is believed to say a mediation led to an agreed cost of £29.1m for the first phase of the project.
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When the contract for the work was initially agreed in 2011, the local authority estimated it would cost around £17.8m.
A total of £19.9m is now left of the budget for the whole project.
A report on phase two of the flood prevention work will go before the council’s transport and environment committee in June, with work on the project to be finished by the end of August.
Councillor Alasdair Rankin, said: "Phase One of the Water of Leith Flood Prevention Scheme, which will protect hundreds of homes from the risk of flooding, is now nearing completion.
"The mediation process has come to a conclusion, enabling the scheme to move forward, with cost and time certainty. Revised governance arrangements are in place to ensure that the problems that affected the first phase are not encountered during the remainder of the project."
(JP/CD)
