Great question! One of the earliest papers (to my knowledge) proposing such a test was by Haas and Ross via the Lens-Thirring effect. Aside: note that the 'gravitomagnetic' field doesn't actually involve magnetism per se.
Lorenzo Iorio reports a possible marginal detection of this effect. I'm not wise enough to comment on its validity/credibility, but I can say that many in the field are skeptical. This additional article might also be helpful.