yes it is possible as I have done it myself, but it does require several file changes
in global.php
find:
$privatemessage_text = $lang->sprintf($lang->newpm_notice_one, $user_text, $pm['pmid'], htmlspecialchars_uni($pm['subject']));
replace:
$privatemessage_text = $lang->sprintf($lang->newpm_notice_one, $user_text, $mybb->settings['bburl'], $pm['pmid'], htmlspecialchars_uni($pm['subject']));
find:
$privatemessage_text = $lang->sprintf($lang->newpm_notice_multiple, $mybb->user['pms_unread'], $user_text, $pm['pmid'], htmlspecialchars_uni($pm['subject']));
replace:
$privatemessage_text = $lang->sprintf($lang->newpm_notice_multiple, $mybb->user['pms_unread'], $user_text, $mybb->settings['bburl'], $pm['pmid'], htmlspecialchars_uni($pm['subject']));
in inc/languages/<language>/global.lang.php (do this for all your supported languages)
find:
$l['newpm_notice_multiple'] = "<strong>You have {1} unread private messages.</strong> The most recent is from {2} titled <a href=\"private.php?action=read&pmid={3}\" style=\"font-weight: bold;\">{4}</a>";
replace:
$l['newpm_notice_multiple'] = "<strong>You have {1} unread private messages.</strong> The most recent is from {2} titled <a href=\"{3}/private.php?action=read&pmid={4}\" style=\"font-weight: bold;\">{5}</a>";
find:
$l['newpm_notice_one'] = "<strong>You have one unread private message</strong> from {1} titled <a href=\"private.php?action=read&pmid={2}\" style=\"font-weight: bold;\">{3}</a>";
replace:
$l['newpm_notice_one'] = "<strong>You have one unread private message</strong> from {1} titled <a href=\"{2}/private.php?action=read&pmid={3}\" style=\"font-weight: bold;\">{4}</a>";
Template changes:
in global_pm_alert you need to insert $mybb->settings['bburl'] into the image tags
<div class="pm_alert" id="pm_notice">
<div class="float_right"><a href="{$mybb->settings['bburl']}/private.php?action=dismiss_notice&my_post_key={$mybb->post_code}" title="{$lang->dismis_notice}" onclick="return MyBB.dismissPMNotice()"><img src="{$mybb->settings['bburl']}/{$theme['imgdir']}/dismiss_notice.gif" alt="{$lang->dismis_notice}" title="[x]" /></a></div>
<div>{$privatemessage_text}</div>
</div>