In an interview with ABC News, Mr Obama said that Mr Trump must “respect” US institutions.
“After you have been sworn in,” he said, “you are now in charge of the largest organisation on Earth”.
He warned that there was a difference between governing and campaigning.
“There are world capitals and financial markets and people all around the world who take really seriously what he [Mr Trump] says,” Mr Obama said.
Mr Obama also talked about the US intelligence agency’s report into alleged cyber-attacks by Russia and the attempt to influence the 2016 US presidential campaign.
He said that he had “underestimated” the impact of such attacks.
“I think that I underestimated the degree to which, in this new information age, it is possible for misinformation… and so forth to have an impact on our open societies.”
He said that a conversation had taken place with Mr Trump in which he had discussed the importance of having faith in the intelligence community.
“There are going to be times where the only way you can make a good decision is if you have confidence that the process is working,” he said.
Last week Mr Trump said he was a “big fan” of intelligence agencies, after months of casting doubt on the Russian link to the security breach. But he later raised questions over how the Democratic Party had responded to the cyber-attacks.
“How and why are they so sure about hacking if they never even requested an examination of the computer servers? What is going on?” Mr Trump asked in a tweet.
Mr Trump will be inaugurated on 20 January.