Suckering is very characteristic of cottonwoods because the root systems are vigorous and shallow. The suckers emerge to create additional plants, so letting the suckers grow will give you more cottonwood trees in unwanted areas.

The growth regulator product SUCKER-STOPPER® RTU may be only marginally helpful, as it is not listed on the label as being an effective treatment for the genus Populus (cottonwoods, aspens, poplars). Avoid herbicides. Since the root system is interconnected, the herbicide will move into the existing tree causing much damage.

Digging to remove the roots can also harm the tree, and will not be a long-term solution as the tree will continue to sucker in many other places.

Others decide to just remove cottonwood trees, however, the root system does remain alive for up to several years with continued suckering.

Unfortunately, there isn't anything you can do other than prune the suckers off at ground level. The other alternative is opting for a patio hardscape that is more solid and slab-like where suckers cannot come up through it. There just isn't an easy answer for this problem, especially where the desire is to not harm the tree.