In a previous paper, we demonstrated that the linearized general relativity could explain dark matter (the rotation speed of galaxies, the rotation speed of dwarf satellite galaxies, the movement in a plane of dwarf satellite galaxies, the decreasing quantity of dark matter with the distance to the center of galaxies’ cluster, the expected quantity of dark matter inside galaxies and the expected experimental values of parameters Ω_dm of dark matter measured in CMB). It leads, compared with Newtonian gravitation, to taking in account the second component (gravitational field) of the gravitation (imposed by general relativity) without changing the gravity field (also known as gravitomagnetism). In this explanation, dark matter would be a uniform gravitational field that embeds some very large areas of the universe generated by the clusters. In this article we are going to see that this specific gravitational field, despite its weakness, could be soon detectable, allowing testing this explanation of dark matter. It should generate a slight discrepancy in the expected measure of the Lense-Thirring effect of the Earth. In this theoretical frame, the Lense-Thirring effect of the “dark matter” would be a value between around 0.3 milliarcsecond/year and 0.6 milliarcsecond/year in the best case. In the LAGEOS or Gravity Probe B experiments, there was not enough precision (around 0.3% for the expected 6606 mas·y^﹣1 geodetic and around 19% for the expected 39 mas·y^﹣1 frame-dragging precessions). In the GINGER experiment, there could be enough one; the expected accuracy would be around 1%. If this discrepancy was verified, it would be the first direct measure of the dark matter.