From Glauber calculations of the absorption cross section of nucleons on air (${\ensuremath{\sigma}}_{\mathrm{abs}}$) we find that present extensive-air-shower data cannot resolve the question of the asymptotic energy behavior of the nucleon-nucleon total cross section [${\ensuremath{\sigma}}_{t}(\mathrm{NN})$]. Relative differences between various asymptotic extrapolations for ${\ensuremath{\sigma}}_{t}(\mathrm{NN})$ are reduced by about a factor of 3 in ${\ensuremath{\sigma}}_{\mathrm{abs}}$ through the Glauber conversion. Novel techniques or conventional experiments with a better determination of electron and muon numbers in extensive air showers will be required in order to obtain useful information on proton-proton cross sections in the asymptotic region.