George HW Bush and his son George W made the criticisms in an upcoming book
George HW Bush has described the current US president as a "blowhard" who is driven by "a certain ego."
Mr Bush, who voted for Hillary Clinton in last year's election, reportedly made the criticisms in historian Mark Updegrove's coming book The Last Republicans.
In the book, he and his son George W talk about why they think their Republican successor is not fit for office.
According to reports, Mr Bush Snr told the author in May last year: "I don't like (Trump).”
"I don't know much about him but I know he's a blowhard,” he said.
"And I'm not too excited about him being a leader."
Mr Bush Snr, 93, and his son, 71, were the 41st and 43rd presidents of the US.
Both were Republicans, with Democrat Bill Clinton taking office in the period between their presidencies.
The book follows a speech by George W Bush in which he warned that bigotry, white supremacy and falsehoods were coarsening the tone of the US and threatening its democracy.
In the speech in New York last month, Mr Bush Jr did not mention the president by name but he was clearly criticising the current administration and its policies.
Mr Bush Jr left his ballot blank in the election and said in the book that Mr Trump "doesn't know what it means to be president."
He said he President Trump’s election had surprised him, also expressing surprise in his lack of humility and inability to recognise his own limitations.
He said: "As you know from looking at my family, (humility) is a certain heritage, that's what they expect and we're not seeing that (in Trump).”
He also said Mr Trump had failed to surround himself with more knowledgeable people, a key part of being president.
After Mr Trump had described himself as his own adviser, Mr Bush had said: "Wow, this guy really doesn't understand the job of president."
Mr Updegrove said the younger Bush had inspired the title of his book, sharing a fear before the election that he would be "the last Republican president."
White House spokeswoman Sarah Sanders said in a statement: "The American people voted to elect an outsider who is capable of implementing real, positive, and needed change - instead of a lifelong politician beholden to special interests.
"If they were interested in continuing decades of costly mistakes, another establishment politician more concerned with putting politics over people would have won."
Mr Bush Jr also refuted claims that his vice president Dick Cheney and defence secretary Donald Rumsfeld had controlled the White House from behind the scenes, telling Mr Updegrove that they "didn't make one f*****g decision."