The midfielder, 33, joined Vale on non-contract terms back in August having been released by Blackpool at the end of last season.
He agreed to play for free in a bid to impress but with a deal put to him in the last 24 hours, he has decided to turn it down and look for more suitable offers elsewhere.
Manager Micky Adams told the club's official website: "We put an offer on the table, which unfortunately Claus has decided to turn down.
"We are obviously disappointed but he is now considering other options and there is nothing we can do about that except wish him luck."
Jorgensen made five substitute appearances for the League Two club during his month-long stay.