The short answer, as of Git 2.28 at least, is "no".  As Brent Faust wrote, you must set the upstream of the current branch, then run git status, then set it again and run it again, if you want git status to print this information for multiple upstream values.
All is not lost
While you can't get git status to do this, you can use a different shell command to do what you want:
counts=$(git rev-list --count --left-right $chosen_upstream...$branch)
# note: three dots
The counts variable now contains two values: the "remote ahead, me behind" value, and the "remote behind, me ahead" value.  If both values are zero, your branch and the chosen upstream are even.  (If you want the counts swapped, swap the $chosen_upstream and $branch variables.)
To turn this into a more-useful shell function (valid in plain old sh and bash both):
# report: invoke as report upstream [branch]
report() {
    local branch upstream count
    case $# in
    1) branch=$(git symbolic-ref HEAD 2>/dev/null) || return 1;;
    2) branch="$2";;
    *) echo "usage: report <upstream> [<branch>]" 1>&2; return 1;;
    esac
    upstream="$1"
    count=$(git rev-list --count --left-right "$upstream...$branch") || return 1
    set -- $count
    case $1,$2 in
    0,0) echo "Your branch is up-to-date with $upstream";;
    0,*) echo "Your branch is $2 commits ahead of $upstream";;
    *,0) echo "Your branch is $1 commits behind $upstream";;
    *)   echo "Your branch and $upstream have diverged,"
         echo "and have $2 and $1 different commits each, respectively.";;
    esac
}
(The output from the above is designed to match that from git status, and isn't really appropriate for the two-argument form, but it shows how to do what you might want to do here.)
(answered in 2020 due to link from How do I do git status upstream?)