Fire departments do not recommend running cords under rugs, but if it is in an area that is rarely walked on... I have one downstairs under a rug. I bought a thick rug pad, cut in half to have a "path" for the cord. Can't tell the cord runs under the rugs since it is in the channel between the two rug pads.

Some years ago I needed to do this and purchased a special padded cord holder designed to be able to walk on/over electric cords.

Something like this one on Amazon.

I have a cord under a rug in my living room. We have a slab foundation and tile flooring and don't feel like having it jack hammered for an outlet. I put a sea grass rug runner against the wall behind furniture and got the flattest heavy duty extension cord I could find. We use the runner as a path to the master, but not used much by guests so it works well.

Hmmm....maybe that cord cover would work. Might not be too conspicuous on beige carpet.

I have two cords in this very room that run across the top of the area rug. If it weren't for the area rug, I'd have hubby install a floor outlet like we have planned in the library where the rug doesn't go completely under his desk. That's not the case here. I'm not thrilled, but since we rarely have guests, everyone here knows to step over them. They are not in a typical walking pathway for us humans. The cats walk over the often and so far none of them have tripped. ;^)
For your situation, I like the idea of the cord cover that Gramarows posted, but would lay it under an area rug if possible. I like Allison's idea of a thick pad cut to go on either side of it too.

trip hazard, fire hazard, better to do the hard cord cover which can go over or under a rug. I remember quite awhile ago, they were working on wireless electricity...I wonder whatever happened to that?