A number of names have already been linked with the high-profile job, including Patrick Vieira, Massimiliano Allegri and Thomas Tuchel, while Rodgers - a Premier League runner-up with Liverpool in 2013-14 - is also rumoured to be a contender.
Desmond would reluctantly allow the Ulsterman to discuss terms with Arsenal should that interest become solid, with Rodgers enhancing his reputation during two years in Scotland.
"He's an outstanding person and we wouldn't like him to leave, but we can't force anyone to stay," he told Sky Sports News. "Hopefully his love for the club and the setup there and induce him to stay."
When asked whether he would allow Rodgers to speak with Arsenal, Desmond added: "Absolutely. You can't put handcuffs on anybody if they want to go to a club like Arsenal. It will be Brendan's decision and Brendan's decision only."
Rodgers remains on course to win a clean sweep of domestic honours this season, matching the success of his maiden campaign at Celtic Park.