I remember that exact story too! I don't remember the ending but it really freaked me out, even if I did overthink it when I was a kid.
The version I remember seemed to announce where it was each night. Would changing where he slept mean the thing would have to start shambling over to this new sleeping spot or if he suddenly started sleeping downstairs would the thing be able to instantaneously backtrack? And how exactly did that all work since the kid would have to exit his bedroom and go downstairs sometime in the day yet the thing only made progress one step a night?
I actually just Googled and there are lots of little variations on this story. Apparently some of them make it take place all in one night which I find a little less effective. And even less plausible for whatever reason, that there'd be such a talkative unseen malevolent force since it'd probably take more time to announce each step then to move onto the next one... And some of them featuring a neglected doll or something and while others--rather confusingly-- featuring a kid sent to go buy liver but screws up and instead decides to dig up a corpse and harvest an organ(!) and forgetting to rebury the body which is then pissed off at him.
This added element of near accidental-cannibalism pretty much totally kills the story for me. Aside from the fact the ending is a bit of a let down in every iteration since it really does work much better on that impending dread front than actual climax. There was even one Doctor Who episode that managed to build up this same sense of dreaded anticipation in one scene and about four lines. (I forget exactly but it was a bodyless mysterious voice saying something along the lines of "Don't turn around. If you look at me you'll die... I'm right behind you...")