We don't know of any evolutionary track that would allow a star like our sun to develop a black hole inside it through gravitational collapse.
It's conceivable that the sun could capture a black hole that came from somewhere else. For instance, it's possible that there are primordial black holes that were formed in the big bang. Depending on the velocity and size of such a black hole, it might pass through the sun without interacting much, and therefore not get captured.
Assuming that such a black hole did get captured, the sun's normal matter would start accreting onto it. For a very small black hole, this process could be quite slow. There are hydrodynamic limits on how fast matter can be funneled into a small space, and a microscopic black hole is very small in radius, so the "funnel" can be very small. I haven't done the calculation, but one should be able to put an upper limit on how large such a black hole can be, based on the fact that geological records don't show the sun as having changed its luminosity dramatically.
For a description of this type of physics, see Giddings and Mangano, "Astrophysical implications of hypothetical stable TeV-scale black holes," https://arxiv.org/abs/0806.3381 . (This is a discussion of doomsday scenarios for the LHC, and because the doomsday scenarios are very easy to rule out based on standard physics, they consider some very nonstandard possibilities such as large extra dimensions.)