Mr Bush, en route to the G8 summit in Germany, is visiting the Czech Republic and then Poland - two countries where the US seeks to build a missile shield.
Russian President Vladimir Putin has threatened to aim weapons at Europe if the US sets up a missile shield.
Correspondents say the rhetoric has echoes of Cold War rivalry.
White House officials insist that President Bush's speech is not about Russia but rather renewing his commitment to spreading freedom and democracy.
See map of US missile defence bases
However, the BBC's Jonathan Beale, who is travelling with Mr Bush, says the president is expected to talk about the difficulties of promoting democracy in countries such as Russia and highlight concerns about the Kremlin's tightening grip on power.
Washington has dismissed talk of a new Cold War but says the rhetoric coming out of the Kremlin has worrying echoes belonging to that era and wants the words toned down.
Any criticism will not be welcomed by Moscow and can only lead to further strains, our correspondent says.
Washington wants to deploy interceptor rockets in Poland and a radar base in the Czech Republic to counter what it describes as a potential threat from "rogue states" such as Iran and North Korea.
On Sunday President Putin said Iran was not a threat to the US, hinting that Russia was the target.