Rooney previously stated he would stay with the troubled side until the end of the campaign, but recently said protracted efforts by administrators to sell the club could see him leave.
His work at the Rams saw him approached for the Everton job in January.
Kirchner said on Twitter that Rooney "told me he's staying".
In a question and answer on social media, the American businessman added: "We're going to do this together. Excited to rebuild this with him."
Following Kirchner's confirmation as the preferred bidder to buy the relegation-threatened Championship side, Rooney said he "can work on my plans now and see where Chris wants to go".
"We can work on that together," Rooney said last week.
"I love this place and the staff here. The players and fans have been brilliant to me as well."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!