I fixed this in the following way.
Go to the “Network admin”. Then Sites → Edit → Settings → Permalink Structure. Remove blog from it.
Then I removed blog from two lines in table which obtained as a result of the query:
SELECT * FROM wp_options WHERE option_name = "permalink_structure" OR option_name = "rewrite_rules"
Useful link on subject: How To Remove The /blog/ slug From The Permalinks(URL) in WordPress Multisite Installation Without A Plugin
This works with WP 3.8, I have not tried with 3.8.1
Note if you go to the main sites Setting -> Permalink you will still see blog, but it does not applied any more. But if you press “Save changes”, blog slug will be used again. You need to redo previous actions again