Are there any known asteroids with average density similar to that of Earth's?
Apparently 21 Lutetia (~80x120 km) is one of the most dense asteroids known at about 3.4 g/cm$^3$. Unfortunately that is barely over 60% of Earth's density. 21 Lutetia is unique in that scientists believe that it has a primordial core, that is, it has a core from the formation of the solar system (and it hasn't been destroyed after all these years). We believe it is hidden under the surface. It would be rich in metals, explaining the density, and because it's hidden under the surface, 21 Lutetia would look no different from any other asteroid, which is the case. This asteroid is a prime target for mining because of its metal core. I suppose the answer to your question would be no, unless 3.4 is sufficient for your definition of "close".