GoKawiilThe Wall Street Journal first reported the email on Thursday.
Ingle also said that all federal employees are subject to government ethics guidelines that prohibit the use of insider information for financial gain.
"The only special interest that will ever guide President Trump is the best interest of the American people," he added.
The BBC has contacted Kalshi and Polymarket for comment.
Polymarket came under scrutiny in January after a gambler made nearly half a million dollars on the capture of Venezuelan president Nicolás Maduro just before it was officially announced.
It was unclear who placed the bet. The anonymous account had a blockchain identifier of letters and numbers.
The incident raised concerns about whether they had benefited from inside information of the US military operation.
Predictions markets, which host more than $44bn (£33bn) in trades, have become increasingly popular in the past year.
The predictions can be related to anything. They mostly involve sports but users can also place bets on whether, for instance, the US central bank will cut rates or the results of local elections.