The Celebrity Big Brother contestant asked the 'Call My Name' singer on Twitter if she would like to meet him for a romantic rendezvous in Essex.
"@CherylCole how bout u let me take you out on a date a show you how #Essex does it? ;-)," he wrote. "Hahaha! Go ooon u no u wanna #Begging looool."
Norcross had said of his love life shortly before asking Cole out: "If twitter had a relationship status option mine would say, 'Erm....I aint got a f*cking clue' ;-) lol."
The reality star was last month treated for smoke inhalation after fighting a fire at the house of his father Mick.
Norcross is the second Essex-born celebrity to proposition Cole on Twitter after Olly Murs, who asked the former X Factor judge for her phone number.
The 'Dance With Me Tonight' star later admitted that Cole had ignored his tweet and claimed that his request "wasn't anything serious".