That leaves the Rams without a punter listed on the roster for a Week 2 preseason game against the Las Vegas Raiders on Saturday night at SoFi Stadium.
The Rams did not immediately announce who could serve as a reserve punter.
For a fourth consecutive year, Rams coach Sean McVay has opted not to play starters in preseason action, and this year he has taken it a step further, resting several key reserves as well.
A fourth-year pro signed over the offseason, Bojorquez has been challenging the four-time All-Pro Hekker for the position.
Both punted twice in a 13-6 preseason loss to the Los Angeles Chargers in Week 1. Hekker averaged 45.5 yards per punt, and Bojorquez averaged 43.