I don't think the Handlebar has TV's. But the barkeep at Sheffield's in Lakeview (1000 W. School St) agreed to put The Tour on the TV in the back bar, despite lots of Cubs fans wandering the neighborhood with nowhere to go (game was in Arizona). And Sheffield's has an amazing line-up of microbrew taps. Good times.
Versus broadcasts the Tour multiple times each day, padded with interesting other bike racing programming.
i caught some of stage 4 earlier today at Cal's. drink at Cal's! it's at 400 S. Wells, and the bar entrance is on Van Buren. they get versus, and will likely be happy to change the channel to Le Tour (if it's not already on). the four or so bartenders who work there are all riders.