Not without editing the core, which we do not assist with. You'd need to review the image field code found in core.php. The easiest approach is to just edit your CSS responsible for the border and also apply it to the avatar. Example as follows.
FROM:
[code:1]
.padded {
CSS GOES HERE
}
[/code:1]
TO:
[code:1]
.padded,
#mod_login_avatar,
.cbavatar_tr {
CSS GOES HERE
}
[/code:1]
Simply add more selectors to the CSS using commas. The above would affect the profile avatar (table based design only) and the login module as well as anything with the padding class.