According to Eurogamer, three separate sources have confirmed that the Nintendo Switch will have the ability to play GameCube games via Virtual Console. Three games will be available on the GameCube Virtual Console at the Nintendo Switch's launch: Super Mario Sunshine, Luigi's Mansion, and Super Smash Bros. Melee.
As for GameCube titles rumored to be appearing on the Nintendo Switch Virtual Console after the initial three, Animal Crossing is said to be under testing for possible release. Not only is Animal Crossing said to be working on the Nintendo Switch hardware, but the NES games included in the original release are also playable. This means that if Animal Crossing comes to the Nintendo Switch, you'll be playing an emulator on an emulator when accessing the NES games.
GameCube titles on the Virtual Console have been a fan request for years. However, it's likely the Wii U might have lacked the ability to emulate GameCube games correctly. Digital Foundry ran tests using the NVIDIA Tegra X1 chipset, which will be powering the Nintendo Switch, to see how well the Dolphin GameCube/Wii emulator ran. The results conclude that the Nintendo Switch should be able to run GameCube games at least as well as their original state.
Nintendo hasn't commented on these rumors, but whether they're true or not will likely be answered in the Nintendo Switch hands-on event in early January. GameCube gaming on-the-go sounds great, but until we get confirmation, we'll be keeping our fingers crossed.