He amde it clear that the final decision rests with each EU member state during a conference with Finnish Interior Minister Mari Rantanen.
Brunner expressed concern about the increase in visas granted in some nations and announced bilateral talks to assess the need for such measures.
He announced a new migration strategy for the end of the year, focused primarily on combating irregular migration.
Analysts interpret these statements as a reflection of internal divisions within the EU regarding sanctions against Russia.
Russian Foreign Ministry spokeswoman Maria Zakharova previously described such restrictions as a way of locking themselves in their own cage.
European media speculated about the inclusion of stricter measures in the 19th sanctions package, including restrictions on the movement of Russian diplomats.
However, EU sources ruled out an outright ban on tourist entry in the short term.
