Local councillors will be asked to approve the contract award next week.
McAlpine is understood to have submitted the lowest bid for the project to demolish and replace the ramp and create a new street grid between Dundee city centre and the river.
McAlpine’s bid came in at £8.1m – around £1.2m below the highest price bid.
The cost of other related works is expected to see the project cost rise to £11.3m.
The works also include construction of traffic signals, drainage, and utility services.
Ken Guild, Leader of Dundee City Council and Chair of the Dundee Waterfront Board committee said: “Work on the Central Waterfront development is gathering momentum and we are making significant progress.
“We have seen the recent project stages coming in earlier than scheduled and they are also being delivered at a lower cost than we had budgeted for.
“With most of the development sites now having been cleared, the waterfront area is attracting significant interest and there is further investment to come.”
The project is part of the Dundee Waterfront project, which will see the creation of a new grid-iron street pattern, based on historical routes.
The council plans to create a new civic space and construct the £45m V&A Dundee on the prime waterfront site.