The Sunday Mirror says the 48-year-old boss, sacked by Gillingham last month, believes he can succeed Ian Holloway, keep Palace in the top flight and have his contract reviewed next summer.
Allen said: "I honestly believe that with the players Palace have got, I can keep them up.
"If you look at my record, I have been able to turn things around at clubs who looked doomed.
"I would love the chance to manage at Premier League level and I am confident that I can keep Palace in the top flight."
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