The London Road club tentatively asked the Mariners about the 19-year-old last month but were told a seven-figure fee would not prise him away from Blundell Park.
Since then, Grimsby manager Mike Newell and chairman John Fenty have stated that their player of the year would not be leaving the club. But, undeterred, Peterborough made an approach for Bennett on Thursday.
"When we initially asked about Ryan the Grimsby chairman said they wouldn't even sell him for a million, but certain things have changed as they always do in football," director of football Barry Fry told the Peterborough Telegraph.
"We put in another bid for him yesterday (Thursday) and are awaiting a response."
Fenty has now delivered that response and told the Grimsby Telegraph: "It's a derisory bid and the club will not be accepting it at all."