Mr Barilaro's controversial appointment to the coveted role has plagued the government for months and prompted a review by former Public Service commissioner Graeme Head, which recommended an overhaul of the NSW ministerial code of conduct.
Liberal MP Stuart Ayres resigned as trade minister earlier this month after a draft from the review raised questions about whether the recruitment process for the $500,000 a year New York-based role was at made at arms length from government.
Mr Kean was grilled at NSW budget estimates on Monday about a text exchange he had with Mr Barilaro in the month the job was formally advertised.
Labor's Daniel Mookhey put to Mr Kean that with the job advertised in late December 2021 "if you were having conversations with John Barlilaro before that ... then he was accessing senior members of the government before it was publicly advertised".
"When precisely were you texting Mr Barilaro about the New York job?" Mr Mookhey asked.
"At some stage during December. I can't be sure when that was," Mr Kean replied.
"It was a text I responded with something supportive and I didn't give it a second thought."
He said he received a lot of "strange and wacky requests" from Mr Barilaro.
"He texted me that he wanted to build nuclear power plants all over the state. He texted me that he wanted to go to the federal parliament. He texted that he wanted to take the National Party out of the government. He texted me that he wanted to kill koalas," Mr Kean said, later retracting the koala comment.
"John Barilaro made lots of outlandish claims that weren't genuine."
Mr Barilaro flagged he was leaving politics in October 2021 and was appointed to the trade role in June this year, before stepping away from it two weeks later, saying the position had become untenable.
A separate parliamentary inquiry is investigating the appointment and its terms of reference have now expanded to look at all trade commissioner roles, with a focus on the agent-general position in London.
Premier Dominic Perrottet said he was yet to receive advice on whether he could still be confident in the London agent-general Stephen Cartwright, following the release of documents revealing he was not the highest-ranked candidate by the recruitment firm NGS Global.
"I'm waiting to receive advice from the minister in relation to that matter," he said on Monday.
"There's obviously matters at the moment the minister for trade (Alister Henskens) is looking at with respect to expenses," he said told reporters.
The future of Investment NSW chief executive Amy Brown is also uncertain following the Head inquiry.
Mr Perrottet declined to say if he had received advice on her future, saying it was a matter for the Department of Premier and Cabinet secretary Michael Coutts-Trotter.
"He's working through that now," Mr Perrottet said.
Ms Brown is expected to face budget estimates next week if she remains in the role.
