The prospect of bringing an asteroid back to the vicinity of the Earth is tantalizing. Upon retrieving an asteroid and returning it to the Earth-Moon system, astronauts will have the opportunity to study an asteroid in close proximity. This paper explores the orbital dynamics of an asteroid in orbit around the Earth–Moon libration point, EML1. The dynamics of the motions for a spacecraft in close proximity to an asteroid are found using the circular restricted four-body problem (CR4BP). Treating the problem as the superposition of two circular restricted three-body problems (CR3BP), the asteroid becomes an additional gravitational perturbation to a spacecraft close to the asteroid. A trade study to examine the near-term behavior of a spacecraft’s orbit relative to the asteroid is conducted via a series of simulations utilizing a variety of variables such as the asteroid’s location and mass relative to the spacecraft, the size of orbit, and the varying of initial conditions. The characteristics being studied are the stability of the spacecraft’s orbit over a short duration as well as the range between the spacecraft and its target asteroid over the mission duration. In addition, the minimum safe stand-off distance between both objects is determined to ensure no collisions or orbital instability. The ultimate goal is to obtain various datasets to deduce the most stable conditions for placing an asteroid and to determine where to fly a spacecraft in formation with the asteroid in orbit about the EML1 libration point. The results show the viability of certain orbital parameters as well as the prevalence of impacts. These datasets can be useful in the design of potential future missions.