Contractor Carillion is claiming compensation from Edinburgh City Council after the Niddrie Burn Restoration Project was completed eight months late last year.
It is understood the builder wants the settlement for costs it incurred due to the over-run, which was caused by bad weather.
The council said the £11m project came in on budget -although some aspects such as a bridge were mothballed - and insists it is not legally liable to provide compensation.
The project saw a mile of the Niddrie Burn realigned to form a river corridor, with landscaping, footbridges and a flood storage area in parkland.
The storm drain that runs half a mile through parkland for has literally divided a community by cutting in half Hunters Hall Park. A bridge is still planned but the timescale is unclear.
David Walker, Labour councillor for Craigmillar, said: "I'm disappointed to hear that Carillon will be making a claim against the council for additional costs. This project is behind schedule, only partially completed and many of the environmental improvements that the community were promised have still to be delivered.
"Any payment to Carillion is likely to have a negative impact on the council's budget and cause further delays to phase two."
Paul Nolan, of Craigmillar Community Council, said the area was now surrounded by a "large medieval-style moat" that has divided the park for the first time in 40 years.
He also claimed it had not delivered any of the environmental benefits promised.
Council papers reveal "the contractor intimated a potential claim" and "external technical and legal advice is currently being sought".
Separate documents revealed: "Some elements would have benefited from formalised reporting, planning at the outset and a greater emphasis on detail, for example governance reporting arrangements, contingency planning and risk analysis. The project needs to resolve a number of compensation events valued at £1.3m which remain in dispute before it can close completely."
The council has already been locked in a series of disputes blamed for helping push the city's controversial trams bill up from £545m to £776m. Another contractual issue emerged during the building of the Water of Leith Defence system, which helped push up the original £11.5 million bill to £21 million.
A spokesman for Carillion said: "We can confirm that we are in discussions with Edinburgh City Council regarding some issues which have arisen relating to the Niddrie Burn Restoration Project. Relations with Edinburgh City Council remain positive."
A council spokesman said: "The council received a compensation claim from Carillion. We considered this and responded."