This is a more complicated question that you probably realise.
This first point to make is that the speed of light is always locally $c$, that is, if you measure the speed of light at your location you will always get the result $c$. The problem comes when you measure the speed of light at some location distant from you.
To measure the speed of light locally we use a coordinate system and measure time and position to locate spacetime points $(t, x, y, z)$. Locally the coordinate system is just the good old flat space coordinates as plotted on graph paper by generations of school children studying physics, and we determine the speed of light or anything else by measuring $dx/dt$ (assuming the object is moving in the $x$ direction). The problem comes when we extend our coordinate system, for example near to the event horizon of a black hole. If we monitor a light ray heading towards a black hole then our coordinates won't match the coordinates of another scientist hovering near the event horizon. That means if we calculate $dr/dt$ to get the light speed we will get a different (slower) value than the other scientist.
But does this mean the speed of light is really less than $c$? I think it depends on what you mean by the word really. For example suppose we give that other scientist a mirror and measure the time for the light to be reflected back to us. If we divide twice the distance to the other scientist by the time then we'll get a velocity of less than $c$. But it could be argued that this is because the distance to the other scientist is actually greater than we think it is, so the light travelled farther than we think and it is still travelling at $c$. There is some discussion of this in The bigger the mass, the more time slows down. Why is this?.
Which is all very well, but doesn't answer your question. I think most physicists wouldn't pay to much attention to anything that is coordinate dependant, and wouldn't regard the $dx/dt$ calculated using your coordinates as especially physically significant.