An offshore system with a rig and a cantilever having a rigid cantilever structure with an operational end. The rig is provided with a first support rail which extends along a side of the deck supporting structure. Travelling and swivelling assemblies are arranged between the first support rail and the cantilever beams. The rig is provided with one or more second support rails that extend over the deck in a direction transverse to the first support rail. The system has a cantilever inner end carrier device travels over and engages on said one or more second support rails. This device is connected via a vertical pivot axis swivel to a connector part of the rigid cantilever structure at or near its inner end to provide a rotation axis at a fixed location relative to the structure for the rotational motion of the cantilever, the cantilever inner end carrier device holding the cantilever inner end relative to the deck.