Time loops should be possible if time travel is possible – and I think time travel could be possible, in theory. So, yes.
There’s an episode of “Star Trek: The Next Generation” called “Cause and Effect” that I really like, where they’re stuck in a time loop, at the end of which their ship keeps getting destroyed and that sends them back in time again (with no memory of what happened), until they eventually figure out what’s going on (because they get that feeling of “deja vu”) and are able to change it. It’s a bit like “Groundhog Day” in space, I suppose!
In theory, I think so – there’s something in the theory of general relativity called a “closed timelike loop” which would do this. I don’t properly understand it though, and certainly no-one has ever seen one. They’re a fun device in science fiction though!