Posh lodged an improved bid for the youngster at the end of last week after making what Grimsby chairman John Fenty described as a 'derisory' initial offer.
Despite the "substantial" sum now on the table for Bennett, Fenty believes the club's standpoint will be to keep hold of the promising 19-year-old.
"I have spoken to the manager and staff at the club and our mindset is that we probably won't be accepting this second bid," Fenty told the Grimsby Telegraph. "However, I must stress at this stage that we have ruled nothing out or in - we have not turned the bid down yet.
"The second bid was a substantial increase on Peterborough's first offer and it is difficult for a club like ours to refuse such a substantial amount that would probably break records at this level. But we recognise Ryan's potential and we have other clubs like Scunthorpe and Blackpool showing an interest and at least two other high-profile sides at that level and above also keeping an eye on the situation.
"One of those has confirmed their interest in writing so they may make a move as well. As things stand, we are erring on the side of keeping hold of Ryan."