Paul Dickov resigned following Latics' 3-1 defeat to Walsall at the weekend, an eighth defeat in nine matches that leaves the club one point above the League One relegation zone.
"The phone's been ringing off the hook, there's been lots of enquiries," Corney told BBC Radio Manchester on Sunday.
"We've got to try and find the right one. The hardest part of is to find the manager.
"The only good news is I can share it with Notts County because they also got rid of their manager (Keith Curle), so there's a couple of options out there at the moment.
"I've probably already had 25 to 30 applications, it's just incredible. That's via text, via agents, emails.
"That's the nature of football. We're not the biggest club but we are a club in the Football League, there's only 72 of us and the jobs are coveted.
"We're always going to get a lot of applications. I heard the other day there was a club that got rid of their manager in the division below and they had 150 applications. It doesn't surprise me at all."
Oldham host Yeovil on Tuesday and Corney concedes he and his board have no margin for error in appointing Dickov's successor with the games coming thick and fast.
"It's really a case of sitting down with my board and evaluating the candidates," he added. "Obviously you want to do it quickly because we have a game (on Tuesday). At the same time you don't want to make mistakes."