The 43rd President said it was important for a free press to hold power to account, and that lecturing other countries on freedom of expression would become more difficult if "we're not willing to have one ourselves".
Asked on NBC's Todayprogramme whether, as he faced harsh criticism during his time in office, he ever considered the media "to be the enemy of the American people", he said: "I consider the media to be indispensable to democracy.
"We need an independent media to hold people like me to account. Power can be very addictive, and it can be corrosive, and it's important for the media to call to account people who abuse their power, whether it be here or elsewhere.
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"One of the things I spent a lot of time doing was trying to convince a person like Vladimir Putin, for example, to accept the notion of an independent press. And it's kind of hard to tell others to have an independent, free press when we're not willing to have one ourselves."
Mr Bush was also asked whether President Trump's promises to unify the US had been borne out in his words and actions.
He said: "First of all there's only been one month in office. He's got four years. Secondly, I think you have to take the man for his word that he wants to unify the country and we'll see whether he's able to do so.
"It's hard to unify the country, though, with the news media being so split up. When I was President, you know, you [TV news channels] mattered a lot more, because there was like, three of you.
"And now there's all kinds of information being bombarded out, and people can say things anonymously. It's just a different world."
NBC was one of the sections of the "FAKE NEWS media" Mr Trump denounced as "the enemy of the American People" in a tweet on 17 February.
At the Conservative Political Action Conference last week the President doubled down on his claim. He said: "They are the enemy of the people. Because they have no sources. They just make them up where there are none."
On Twitter Mr Trump has condemned behind-the-scenes leaks from inside the White House that have provided fodder for political reporters in Washington.
He said: "The FBI is totally unable to stop the national security "leakers" that have permeated our government for a long time.
"They can't even find the leakers within the FBI itself. Classified information is being given to media that could have a devastating effect on U.S. FIND NOW".
Mr Bush was also asked whether he agreed with Mr Trump's travel ban. The ban, handed down by executive order, temporarily blocked travellers from seven Muslim-majority countries and suspended entry for Syrian refugees indefinitely.
It was struck down by judges and a new version is expected this week.
Hinting at his position, Mr Bush said: "I am for an immigration policy that is welcoming and upholds the law."
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