You may have to leave them where they are and go with underground drains.
The reason being, that the gutters are pitched slightly toward the downspout side.
So if you move the leader location and plug the old hole, you'll end up with water sitting in the low end of the gutter.
If the underground pipe job is done correctly you wont have any problems. That, and put catch screens inside the gutter, right over the downspout, if you get a lot of leaves.