Joe Biden's speech in Poland has attracted much attention due to some of his off-cuff remarks (see, e.g., the discussion here). I however found curious another passage:

A dictator bent on rebuilding an empire will never erase a people’s love for liberty. Brutality will never grind down their will to be free. Ukraine will never be a victory for Russia — for free people refuse to live in a world of hopelessness and darkness.

Essentially, President Biden taps here into a popular belief that Putin is aiming at restoring the Soviet empire (although he leaves it deniable, as words empire, dictator do not appear in the same sentence with Russia).

To my knwoledge there is little factual support for this belief. One often cites Putin's quotes about the USSR, such as

The demise of the Soviet Union was the greatest geopolitical catastrophe of the century.

OR

Anyone who doesn't regret the passing of the Soviet Union has no heart. Anyone who wants it restored has no brains.

None of them however directly suggests that he wants to restore the USSR, and none of them is an official policy statement. They are more the talk of a politician, appealing to the population in a country, where many people have positive memories of the USSR - indeed many Russians were plunged into poverty and lost livelihood after the collapse of the Soviet Union, and in the 90s nearly a half of them used to support the Communist party, which literally promised the restoration of the Soviet Union (and many still vote for it).

So, what is the claim about the desire of Putin to restore the USSR based upon? (This subjective desire itself is unprovable, but the question is only about the basis for the claim in the headline.)