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The Posh, who last month lost striker Leon McKenzie to Norwich, are confident that Curo will sign the two-and-a-half year deal that has been offered to him.
The London Road side have already had a bid for Cureton accepted by the Busan Icons who currently hold the player's registration.
Boss Barry Fry revealed that he is licking his lips at the prospect of signing such a top quality forward.
Fry told BBC Sport: "It's not often a player of his quality becomes available in these circumstances. He's a proven goal scorer at our level and higher, and he would be a magnificent signing.
"He's very keen to come back to England and we are the only club to have agreed a fee, but we've only got until the weekend before the transfer window closes."
Since Cureton revealed that he wanted to return to the UK back in November a whole host of clubs have made enquiries about the Bristol-born frontman who has repeatedly proved himself in the Second Division.
Sunderland, chasing promotion to the Premiership, took the 28-year-old on trial whilst it was rumoured that Reading were considering re-signing the Busan Icons man. Prospective moves to Bristol City, Barnsley and QPR may have fallen through but that hasn't stopped the former Royals fans favourite from desperately trying to seal his return to the UK.
Indeed, with just days remaining until the close of the transfer window it is highly-likely that Curo will accept Peterborough's offer as the idea of reutrning to the Far East for another tortuous six months is not going to appeal.
The 28-year-old began his career at Norwich City before making the switch to Bristol Rovers. Then, in the summer of 2000 followed a move to Reading were he enjoyed a fine first season at the club, netting thirty-one times. In total the hitman scored 54 times in 124 games for the club.