It is certainly possible but highly unlikely. Typically, asteroids are travelling far too fast for Earth’s gravity to make much of an impact on their trajectories. They normally have more than enough energy to escape Earth’s influence even if they approach very close.
However, if they are of the right mass, are travelling at just the right speed, and manage to miss our planet by just the right amount, they could end up in a stable orbit around Earth. This capture process was probably how Mars got its two small moons, Phobos and Deimos.