Yeah, lots of aural exercises can help keep time. In fact, this is how many children are taught. (ie: Clapping, Tapping, Snapping, etc.) Even the environment can help.
Look at the second hand on a face clock or when your alarm goes off on a digital clock or watch. They have rhythmic patterns.
And I bet this happens (I've experienced it lots) Have you ever ridden in the car and activated the turn signal or seen the headlights or tail-lights blink where they appear to be doing so rhythmically only to go off-beat later? That's more of a psychological/neural phenomenon, but quite common.
Or say you're hear a song on the radio, and the drum parts (kick/snare/hat) sync up to whatever you're looking at, (the car in front of you; the store sign the blinks off to the side).
But like Beags said: Try to internalize the clock. However, you'll get into the "Mechanical vs. Free" (Rule vs. Feeling/Uniformed/Loose) and dare I say: "Real vs. Fake" debates.
Learning in strict time/tempo is where you should start, then gradually proceed to go off-kilter.
Once you get good enough at it, when listening to songs, you'll start to realize how off-beat they are. It's the illusion of them being on beat, that makes them sound good.
I know you probably talking along the lines of performing, but this works very well when composing.
Also, doing such exercises will help you count when there is absolutely no concrete rhythm (ie: drum beat). This will be hard at first, but it helps you to not use an aural crutch (ie: metronome). Not using a crutch serves you better; however, use it until you get to that point.