I am planning to replace an existing ceiling light fixture with a recessed light. To put in the recessed light housing I need to remove the existing housing. But the problem is that I do not see any screws in the housing that I can unscrew. Probably this housing is connected to a joist but I do not have attic access to confirm that, or to reach to those screws.

My questions is that can this housing be removed? If yes then how? The existing housing is made of plastic, do I need to break the plastic and then try to get to the screws that are holding it to the joists? Thanks.

Let me get this correct in my mind. You have a current fixture hanging from your ceiling and you want to remove it a install a recessed fixture, is this correct? If so that old fixture was connected to a box that might be screwed or nailed to a joist or even attached between joists with a metal bracket. Is that box blue plastic by any chance?

You correctly understood the problem. I cannot check the color of the box right now but it is a dark colored box, like black or dark gray. It is circular in shape.

If you want the new fixture to be in the exact location the box would need to come out. As I mentioned earlier it may be attached to a rafter, there can be a plate attached to the back of the box that gets nailed into the rafter above the box. If you have bought your new recessed light check to make sure it is designed for remodeling and not new construction. There may be a template inside the box so you could start by making a new larger opening around the old box but try to determine where the rafter is located so you don't end up with it running through the side of your hole. Back to you.

Like Ballenger is saying, If you determine rafter location, you should be able to remove box, and install 6" can with no ceiling repair. Be great if 1 romex was going to it. Would be a MAJOR problem if you had 6 conduits into a box with an extension.
In that case I would install Light slightly over or where you want, and run 1 wire from box to new can. Then use the round screwless plastic cover, you can hardly see it. Or Smoke Detector at old light location.
Pushing up on the plastic box on the different sides may help determine what side it is nailed to. Then draw 6" remodel can outline, starting at the edge of the rafter and cut out.