Is it possible to use the_ID with offset

some of the below output links, others return a post object, which you can use to build your own links.

in single posts:

next_post_link()
http://codex.wordpress.org/Function_Reference/next_post_link

previous_post_link()
http://codex.wordpress.org/Function_Reference/previous_post_link

get_adjacent_post()
http://codex.wordpress.org/Function_Reference/get_adjacent_post

in the loop:

get_next_post()
http://codex.wordpress.org/Function_Reference/get_next_post

get_previous_post()
http://codex.wordpress.org/Function_Reference/get_previous_post

can you describe in more detail, where you want to get this next post id ( or link )?

if you only want to output the post ID, then try:

<?php $next_post = get_adjacent_post(false, '', false); echo $next_post->ID; ?>

(next post is the one newer than the current)
(similar for previous post)