Rugged defender Branston captained the London Road side during a loan spell last season and has been offered terms by Posh's excentric boss Barry Fry.
But Turner has spoken to the player after the recent testimonial for Paul Hurst at Millmoor and has offered him a more lucrative deal to come to Hillsborough.
He has been honest enough to say that money is a big factor in a possible move destpite showing his admiration for Fry, who gave him a chance last season.
Branston said: "Barry Fry's been great with me while I was there and I was probably close to doing a deal with him, But the money was miles apart to be fair and it doesn't look like I'll be going there now."
He claims he has not had any official approach from Wednesday despite Turner's interest, but believes playing for The Owls will be too good an opportunity to turn down.
He continued: "If any deal can be done then I wont be stupid and mess it up by asking for too much money, I understand the position Wednesday are in financially, but to put the blue and white shirt on would be great."
Wednesday and Rotherham have had quite a few transfer dealings in the last few years with The Owls signing Mark Robins and goalkeeper Matt Clarke while The Millers have taken Adem Poric on loan and gave a chance to Ritchie Barker after his Hillsborough rejection.
Turner has admitted his interest in not only Branston but Bradford’s talented left-back Paul Heckingbottom and former United loan winger, John-Paul McGovern.