Probably not – at least not the kind we should be worried about.
There is very little research on this topic and the studies that exist don't tell us a great deal about the possibility of harmful E. coli being spread by reusable bags.
The idea that reusable shopping bags could harbour dangerous bacteria first came from a University of Arizona study in 2010 that swabbed grocery bags at random in supermarkets in California and Arizona.
The researchers found that 8% of sampled bags tested positive for generic E. coli, which sounds very much like bad news. However, there are 45 known strains of E. coli and generic E. coli is not considered a pathogen (it doesn't cause disease).
Generic E. coli is typically found in our intestines and is considered a healthy part of our gut biome so while it does not spread disease itself, its presence can be indicative of faecal or meat contamination.
The authors of the study did not outline if any of the six strains of E. coli that can harm humans were present.
Then in 2012, a study was published from the University of Pennsylvania that demonstrated a significant rise in E. coli in the San Francisco area after the enforcement of a plastic bag ban.
The authors looked at data of E. coli-related emergency room visits in the San Francisco area as compared to the data from neighbouring Bay Area counties that did not enforce a bag ban.
They noted a jump in visits in San Francisco in the ten quarters following the plastic bag ban and concluded: "Our results suggest that the San Francisco ban led to, conservatively, 5.4 annual additional deaths."
However, the findings were refuted by San Francisco health officer Tomás Aragón, who penned an open letter to the authors and said that their study design did not allow them to make causal links.
Aragón also pointed out that the jump in E. coli cases was not unlike the increase witnessed throughout the United States, Canada, and Europe.
It should also be noted that the University of Arizona study found that machine or hand washing the reusable bags, even without bleach, "was effective in reducing coliform and other bacteria in the bags to levels below detection".