This is a vintage Rex motorbike style bicycle.  Rex Cycles were made in Chicago, IL.  The original Rex Cycles went out of business right after the turn of the century.  The most famous Rex is the 3rd wheel bicycle, on display at the Smithsonian.  Not sure of the connection, but the newer Rex Cycles could have been purchased by Schwinn as one of their many off brands. 
The paint is a blue color.
Headbadge is a nice brass badge with Rex Cycle Works, Chicago on it.

Here's the consensus:  We think that this bike may have been made in the 1930's by Schwinn? under the brand name Rex.  Sometime in the past, this bicycle broke in several places and has been rewelded.  At that time it was painted blue over the clay red color (the clay/red is under the blue in all spots except over the poor welds.  The red color could be a primer, but we doubt that.

Missing:  original wooden rim 28" wheels, any type of accessories, original pedals, seat, rear rack, chainguard?...

***Unrideable, as there is a large crack at the junction of the downtube and the crankhanger (bb).

The 26x1 3/4" steel clincher rims are painted white (Victory style from the 1940's?) and are mated to New Departure hubs (model c coaster brake rear and model M front hub).

Hubs seem correct, but not the size of wheels - large gap between them and the stepped stainless fenders.
Fenders are a little rough, bends, dents, dings abound.

Not much chrome left on the handlebars or the brace.  Not much left of the grips either.

Measurements:

Seat Tube (center to top) - 18.5"

Top Tube (center to center) - 23"

Serial #4 8  7965

Reference #40148