"This is what leadership looks like," Clinton said Thursday on Instagram.
Bush did not win re-election in his 1992 presidential race against the former president but wished Clinton "good luck."
"I wish you great happiness here," Bush wrote on Jan 20, 1993.
Bush, who has not endorsed Trump, communicated great support for Clinton in the letter.
"Your success is now our country's success," he added. "I am rooting hard for you."
After leaving the White House, Clinton and Bush embarked on charitable missions together and even became friends.
Clinton has attacked Trump for implying that he might not accept the results of the election if he loses.
Trump said Thursday he will accept the results of the November election in the event that he wins. Critics have said that this caveat threatens to cast unprecedented doubt from a major party candidate on the legitimacy of the electoral process.
"I would like to promise and pledge to all of my voters and supporters and to all of the people of the United States that I will totally accept the results of this great and historic presidential election, if I win," Trump told supporters in Delaware, Ohio.
The GOP candidate, who is trailing Clinton in both national and key state polls, raised concerns about voter fraud -- something that is extremely rare.
"Of course, I would accept a clear election result, but I would also reserve my right to contest or file a legal challenge in the case of a questionable result," he said. "And always, I will follow and abide by all of the rules and traditions of all of the many candidates who have come before me. Always."