Yes, I would say not impossible, but highly unlikely to happen naturally.
Everything would have to happen perfectly for this to ever happen.
How it could happen:
The asteroid would need to be mostly a metallic variety and shaped and oriented correctly (Blunt body aerodynamic shape faced into the atmosphere, like space capsules during re-entry) to withstand the atmospheric entry and deceleration forces intact. As well as correctly aimed into the atmosphere so as to slow the correct amount, to little and it skips off the atmosphere back into a solar orbit, too much and it impacts earth.
After the initial aerocapture entry into the atmosphere the upper end of the orbit (apoapsis) would need to be oriented such that the moon's gravity would alter the orbit into a more stable configuration (it would need to accelerate the asteroid to raise the periapsis above Earth's atmosphere)
And even after that your interactions with the moon could make your orbit unstable, it would be a very complicated orbital maneuver. We have only performed aero braking on a few space missions, and have never tried an aerocapture maneuver and that is with craft able to supply some maneuverability.