I am buying a new house that has the washer and dryer located on the main floor, but I want to move it into the unfinished basement (as well as add a laundry tub).

There is a rough in for a bathroom directly below where the washer and dryer are currently located, and I'd like to move the washer and dryer next to that rough in (so that the washer and dryer would be in the room next to the bathroom).

Is it possible to use the rough in that exists for the bathroom rough in, or to tap into the existing washer drainage to drain the washer? Or would I have to tear up the concrete to create a new drain?

Do it the shortest possible way. Just take care that the waste water does not has to go up (against gravity). Most machines exhaust pumps cannot lift the water higher than some 60 cm above the top of the machine.

Does the existing drain go straight down into the basement, or after the trap go off horizontal somewhere? If it comes down, and nothing else goes into it, you should be able to cut it off at about the height of the washer. Running the drain up to the old drain may or may work. The pumps are simple centrifugal pumps and their volume drops off as the height goes up. If there was a roughed in drain near by, you could run it horizontal over to behind the washer and up adding a trap. Be very careful the use the more generous drain elbows if you need them. Between threads, hair, and stuff left in pockets, washer drains are prone to problems.

Hi Dan,
If the basement rough in has a lavatory vent already in place then you have 2 options.
You may cut a drainage tee and trap into the lavatory vent, (keeping it close to the floor line) and run up a stand pipe to a few inches higher then the flood rim,(lip) of the washer or couple a stand pipe directly onto the tub stubup. No vent required as the tub's already wet vented through the existing lavatory vent.
I would opt for the lavatory vent as it's inside a future wall line and you can add a short piece to the drainage tee to bring the trap and stand pipe outside the wall. Regards, Tom