Kyrgios lost five straight games from 4-4 in the second set in a match that lasted one hour and 17 minutes. Basilashvili's victory over an under-the-weather del Potro at the weekend was just his second ATP title, the first coming in Hamburg in July.
De Minaur will kick off his Shanghai campaign on Tuesday against Poland's Vasek Pospisil.
Last night, the 23-year-old was dumped out of the Shanghai Masters, following a clash with an umpire who deemed his efforts to be 'borderline'.
It's the third year on the bounce in Shanghai that Kyrgios has departed the tournament in controversial circumstances.
Kyrgios' latest round of unsportsmanlike antics took place Monday night at Qizhong Forest Sports City Arena, where the world's number 38 player suffered a stunning defeat to his 104th-ranked American opponent, Bradley Kahn.
But he also sounded a note of caution.
"I think it's really up to him where he wants to go and what his potential really holds", the 20-time Grand Slam champion, Federer, said. "You have no right to tell me that it's poor", he told the French official.
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Kyrgios was fined after he stormed off mid-way through his first-round exit in the event a year ago, while in 2016 he argued with spectators and was accused of giving away points in another defeat.
The talented but erratic Kyrgios has played in Shanghai four times with his best showing being two second-round finishes.
Added Federer: "He can win bigger tournaments and do all these things". Sanghai is not the only scene of a Kyrgios controversy.
During prolonged exchanges between the pair, Kyrgios was heard saying "I don't care" and then told the umpire: "You have no right to tell me it's poor".
The Swede climbing down from his chair in NY to have a lengthy chat with Kyrgios when he was showing little effort as he trailed by a set and 3-0 down against Pierre-Hugues Herbert.
"I think with Nick when he's fresh he plays great and it's about managing schedules as much as anything, so that he plays no more than three weeks in row and then he gets away from it", Woodbridge said at the launch for next year's Australian Open.