Neil Freeman proposes (hat tip: Blattman) one possible system to bring the House and Senate closer together by making the Senate a far more democratic institution. That is, the Senate was formed in the US Constitution under the federalist ideal that the individual states were themselves sovereign, so senators were chosen by state governments. This brought no end of concern to big-state founders, like Benjamin Franklin, who argued this was no democracy.
So what if the state boundaries were redrawn on occasion - as the House seats are - to create 50 states of roughly equal population (5.6 million)? The thought experiment leads to this:
The main problem he recognizes is that local governments would have to adjust to new state rules.
One of the things I like about it is the re-emergence of a few city-states, an idea I enjoy playing with. Here, New York City is in two (Long Island and New York), Los Angeles, San Francisco, "Orange County," Chicago, and Detroit, plus the greater Dallas and Houston areas. The larger city-states tend to dominate their state's politics, but simply splitting this into New York and (rural) Empire states exacerbates the Senate's rural bias problem. This proposal reduces it significantly.
Of course, it would not be strictly necessary to actually move state boundaries in order to change the notion of "Senate districts" to 50/100 equally-sized districts.
Feasibility? Probably 0. The senators from the lower-population states have a lot to lose.