In theory, yes. In practice, it would likely be highly unstable due to the moon's and the sun's gravity. It would either be flung out of the earth moon system, or crash into one. As for capture, the most likely method is via a gravity slingshot off the moon, it might even enter an orbit that is stable for 100s of years, but that is unlikely. The exception is if it gets caught in one of the Lagrangian Points. L4 and L5 are stable and so could trap an asteroid. There is also evidence that one might be there already.