We designed and fabricated a graded-index (GI) multicore fiber (MCF) compatible with both standard multimode and single-mode fiber for high density optical interconnect application in large-scale data centers. The proposed fiber supports long-distance multimode transmission at 850 nm as well as quasi-single mode transmission at 1310 nm and 1550 nm. The parameters of the GI-MCF have been optimized to obtain both a small differential mode delay at 850 nm and a small mode field diameter mismatch of less than 0.5 μm with single-mode fiber at 1310 nm and 1550 nm with a negligible inter-core crosstalk. In experiment, we successfully realized the multimode operation over 1 km-long GI-MCF at 850 nm and the quasi-single mode operation over 12.4 km-long GI-MCF at 1310 nm and 1550 nm at a data rate of 7×10-Gb/s. The multi-wavelengths multicore transmission was demonstrated for the first time. The experiment results imply that the proposed GI-MCF satisfies various requirements in such as operating wavelength, accessible distance and interconnect density of large-scale data center, and can effectively reduce the fiber numbers and system complexity.