Rodgers is enjoying his time as Celtic manager, but a return to a top Premier League club is something he may find too good to turn down.
Rodgers had a relatively successful tenure at Liverpool, where he almost led the team to the title one season.
Speaking to Sky Sports News, Desmond said: "I'm glad he's the favourite. He's an outstanding person. We wouldn't want him to leave but we won't force him to stay. Hopefully his love for the club and the setup here will enduce him to stay at the club.
"Absolutely [we would allow Arsenal to approach Rodgers]. I don't think you can put handcuffs on anybody if they want to go to a club as good as Arsenal. It will be Brendan's decision and his decision only."
For his part, Arsene Wenger believes Patrick Vieira is the right man to take his job.