Aspas simples devem conseguir o que você deseja.
# " is the special charater
var='"hello "word";'
grep "$var" file
Na página do bash man:
Enclosing characters in single quotes preserves the literal value of each character within the quotes. A single quote may not
occur between single quotes, even when preceded by a backslash.
Enclosing characters in double quotes preserves the literal value of all characters within the quotes, with the exception of $, ‘,
\, and, when history expansion is enabled, !. The characters $ and ‘ retain their special meaning within double quotes. The
backslash retains its special meaning only when followed by one of the following characters: $, ‘, ", \, or <newline>. A double
quote may be quoted within double quotes by preceding it with a backslash. If enabled, history expansion will be performed unless
an ! appearing in double quotes is escaped using a backslash. The backslash preceding the ! is not removed.