The two neighbours have always had complicated relations strained by historical animosity and the Soviet domination after World War II.
A plane crash that killed Poland's president and 95 others in Russia two years ago first brought the nations together, only to push them apart due to disputes over who was responsible.
Around 5,000 Russian fans wanted to march to the stadium on Tuesday and their representatives told Warsaw officials they wanted only to celebrate "the festival of football", the director of Warsaw's security and crisis unit said.
"I've asked them for peaceful behaviour, not to provoke anyone in the streets," Ewa Gawor, in charge of security in Warsaw, told a news conference. "We want this festival to be peaceful. We have had such assurances, nevertheless we will be watchful."
UEFA has told Warsaw to expect around 20,000 Russian fans in the city, spread across the stadium and the fan zone.
A Warsaw police spokesman said they were fully prepared but declined to give precise numbers of how many officers would be on the streets.
Russia coach Dick Advocaat and the country's soccer chief laid a wreath in Warsaw on Sunday to commemorate the victims of the plane crash in an attempt to defuse tensions.
A movement led by Jaroslaw Kaczynski, the ex-president's twin and leader of Poland's main opposition party, questions whether the crash was an accident and says Moscow may be at least partially to blame.
Poles also fear the Russian fans may display Soviet era symbols that remind them of the 44 years under Moscow's domination behind the Iron Curtain.
"Every little thing brings back all the historical grievances which have not been fully resolved," said Andrzej Rychard, a sociologist at the Polish Academy of Science.
Alexander Shprygin, head of the Russian fan association said: "Our walk has nothing to do with politics, fans have nothing to do with politics. All we want is to show support for our team, we do not want any provocations."
A group of Russian and Polish fans also laid wreaths together at a cemetery for Soviet soldiers and at the monument to the 1944 Warsaw uprising, Poland's PAP agency reported.
EARLIER TROUBLES
Russian fans displayed illegal banners and threw fireworks during the team's opening match against the Czech Republic in Wroclaw on Friday and UEFA has launched disciplinary proceedings against the Russian FA (RFS).
The Russian FA has since appealed to its fans to behave.
Beyond isolated incidents, the tournament has so far been mostly calm.
Poland's interior minister said that out of 905,000 fans attending games at stadiums or in the fan zones, only 72 were arrested, including 41 local fans and 10 Russians.
Tensions are also growing between Russian and Ukrainians in the western Ukrainian city of Lviv, which hosts three Group B games. Russian fans scuffled briefly with Ukrain
