I bought a used 2007 Hyundai Tucson 4wd 6 cylinder and it makes a thumping noise like a flat tire. It may go a day or more before it does it. I can visually see nothing loose under it. It happens in 2wd and only when driving. Any ideas on what to check first?

 

Do you know who the previous owner was?
Do you know if it was "gotten rid of " because it requires some expensive repairs?

These questions always come up when problems surface with a recently purchased used vehicle.
Is the condition present when in 4WD? If in both settings check rear driveline u-joints.

Now after that is covered, something like what you describe could be cause by something simple as a defective tire or even a loose wheel. I have had incidents where the wheel was not tightened correctly or fully and the nuts(in this case) were loose. It was fortunate because if left unchecked and corrected, the studs could have been destroyed and the wheel rim ruined and the wheel could have just come off. Please for your own safety and others have that checked ASAP.