The former President also warned that Russia’s Vladimir Putin “is zero-sum. He can’t think, ‘How can we both win?’ He only thinks, ‘How do I win, you lose?’”
Russia has denied trying to influence the November 2016 vote but US intelligence agencies concluded it had.
Mr Trump has previously downplayed the findings but last year said he was “with our agencies, especially as currently constituted with the leadership”.
Mr Bush said it was not clear if Russia’s “meddling” directly affected the election, though he said it was clear they tried and that it was “dangerous for democracy”.
“It’s problematic that a foreign nation is involved in our election,” he added at a conference in Abu Dhabi, and continued: “Putin is a brilliant tactician who has the capacity to detect weakness and exploit it.
“He’s got a chip on his shoulder. The reason he does is because the demise of the Soviet Union troubles him. Therefore, much of his moves [are] to regain Soviet hegemony.”
The Republican leader, who occupied the Oval Office from 2001 to 2009, said Mr Putin “is pushing, constantly pushing, probing weaknesses. That’s why Nato is very important.”
Former FBI director Robert Mueller is currently investigating alleged links between the Kremlin and Mr Trump’s campaign team.
Several people including the Republican’s former national security adviser, Michael Flynn, have been charged.
Mr Trump denies any such links, and has called the investigation a “witch hunt” perpetrated by his political opponents.
The 43rd US President has intervened a number of times to apparently rebuke Mr Trump during the latter’s time in office.
In a book he said “this guy doesn’t know what it means to be President”, and last October, while not mentioning his fellow Republican by name, he said that in recent times “bigotry seems emboldened. Our politics seems more vulnerable to conspiracy theories and outright fabrication”.
Mr Bush broke a longstanding silence in February last year to say that a free press was “indispensable to democracy”, after Mr Trump attacked a number of media companies as “enemies of the people”.
“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”, he added.
At the Abu Dhabi event Mr Bush also said it was important to recognise that the United States had a history of welcoming people regardless of their religion or where they were born, in an apparent criticism of Mr Trump’s policies.
“Our system is broken but we’ve got to fix it. I had tried, but unsuccessfully,” he said without naming Mr Trump. “It is important for our economy and also important for our soul that the immigration system functions well.”
Additional reporting by agencies