Register now for FREE unlimited access to Reuters.com RegisterEU leaders gave mixed messages ahead of a meeting of their ambassadors in Brussels on Monday, with another one planned for Tuesday.
Russia invaded Ukraine - a former Soviet republic that now wants to join the EU - on Feb.24 by air, land and sea.
More than 2,000 people have been detained across Russia for protests at the draft, says independent monitoring group OVD-Info.
On Monday, senior Russian lawmaker Sergei Tsekov told RIA news agency that Russia itself should bar draft-age people from leaving.
Register now for FREE unlimited access to Reuters.com RegisterReporting by John Chalmers, Sabine Siebold, Charlotte Van Campenhout, Writing by Gabriela Baczynska; Editing by Andrew HeavensOur Standards: The Thomson Reuters Trust Principles.