Dortmund are in Russia to face Zenit in the first leg of their last-16 tie.
Dortmund's official supporters group asked fans on social media to "take care of yourselves" in the city.
"There are still ongoing attacks on BVB fans in the city," they said.
"Local supporters are in groups of up to 20 looking for people from Dortmund."
Michael Wollny of Eurosport's German office said that attacks on fans had been confirmed on Dortmund's main fan forum - schwatzgelb.de – and that fans were being advised to stay in their hotels and not travel alone into the city.
Eurosport Russia said that the Russian media have not reported any news of fan attacks.
Zenit have been ordered to close the four sections of the Petrovsky stadium, used by their ultras, for the match and have organised extra security measures following incidents during a group stage clash in Vienna, where their fans displayed racist banners and let off fireworks.
The club's general manager Maxim Mitrofanov said that season ticket holders for the ultras sections would not be allowed in other parts of the stadium, even if they bought tickets.
He also warned of delays entering the ground as club employees checked that none of the banned fans was trying to get in.
"We understand that it's hard to be outside at the stadium for a long time," he told Zenit's website.
"Petrovsky does not have the most comfortable infrastructure, unfortunately, especially in winter time. So we're going to give Borussia fans hot tea, and treat them to pirozhki (Russian pies).
"We'll show them true Russian hospitality, to soften the influence of any cold weather on the match atmosphere."