Barnet were condemned to the National League last weekend, which has led to Allen's departure after a fifth stint at the north London club.
The Bees won five of their eight matches under Allen, who took over in March, but it wasn't enough for them to remain in League Two.
The 52-year-old said in an official statement: "A short while ago I had a meeting with the chairman to let him know that I would not be continuing to manage his and your club.
"It’s been an amazing, unforgettable eight weeks with all the players and the positive energy and the enthusiasm from all the supporters has been quite overwhelming. I can’t thank you enough.
"The chairman has offered me a fantastic contract, offered to increase the budget and had done everything he could, and no blame whatsoever can be put his way.
"For me, it was eight weeks and I am happy, but of course underneath I’m bitterly disappointed and still very upset that we were unable to retain our position in the Football League."
Bookmakers Coral have made Allen their favourite to take over from Allen.
