The tourist attraction on the city's waterfront went to the High Court in 2013 to recover costs to fix problems with the steps, terraces and ceilings.
It said it was necessary to shut the museum from 31 December until the end of February for the "safety and security of collections and visitors".
The £72m museum opened in 2011.
It features Ken Dodd's tickling sticks, the skeleton of the 1900 Grand National winner and rare Beatles artefacts.
The museum said it has attracted 4m visitors since it opened.
The Museum of Liverpool is to close for two months for "essential works" to rectify construction and design issues with the building.