Barnes, who won 79 caps between 1983 and 1995, believes the 37-year-old LA Galaxy midfielder is simply not good enough to make the squad.
"If you look at it from the point of view of winning the Olympics, then Stuart will know what he wants from his team," Barnes told Radio 5 Live.
"From a political point of view given the role he played in bringing the Olympics here, it would be great had he been there.
"But as a coach you have to pick the best squad that will enable you to win and that's why Stuart has made that decision.
"The Olympics is about winning and you're belittling that if you say let's throw a spectacle by picking someone more popular. That belittles the Olympic ideal of excellence.