Speaking in Berlin, Mr Obama said he hoped the US president-elect would “not simply take a realpolitik approach” to dealing with Russia.
He also warned against a cyber arms race, saying there was clear proof Russia had engaged in cyber attacks.
Mr Obama was speaking after talks with the German Chancellor Angela Merkel.
The two close allies stressed the need for continued close co-operation between their countries, as well as between the US and EU as a whole.
Mrs Merkel acknowledged that the US had “shouldered most of the burden” of the Nato alliance – a disparity criticised by Donald Trump – and said European countries “must in future… engage more, so that in the long term the imbalance in defence spending is eradicated”.
Mr Obama said he hoped Mr Trump would seek a constructive relationship with Russia, “finding areas… where our values and interests collide”.
But he added that he hoped Mr Trump was “willing to stand up to Russia when they deviate from our values and international norms”.
This meant, he said, refraining from taking a “realpolitik approach” and cutting deals that could “hurt people or… violate international norms… leave smaller countries vulnerable, or create long-term problems in regions like Syria”.
On the issue of cyber attacks, Mr Obama said there was a difference between “Russian intelligence-gathering” and “meddling with elections or going after private organisations or commercial entities”.
He said he had “delivered a clear and forceful message that…. we’re monitoring it carefully and we will respond appropriately if and when we see this happening”.
He also had this warning for young people: “Do not take for granted our systems of government and our way of life… Democracy is hard work.”