According to Robert Pinder, PhD Microbiology & Animal Science, Purdue University (1996)
Absolutely. If you open a container with glycerol and expose it to air, the various microbes (including bacteria) floating in the air will certainly land on the surface.
Your question is probable better phrased as “do bacteria grow in/on “pure” glycerol to the extent that it can no longer be considered pure / usable?”
my answer then is no. The bacteria would not have a source of nitrogenous compounds to build amino acids, peptides and proteins. As enzymes are proteins and drive the biochemistry in a bacteria, no enzymes mean no biochemical activity which leads to death.
I will slightly differ from Mr. Pankaj. Glycerol not necessarily kill the bacteria, rather mostly they become dormant due to non availability of suitable environment. So, there is always a chance of contamination in glycerol too.