The Posh have already seen three bids for the Scotland U21 international knocked back.
They are refusing to concede defeat in their efforts to bolster their attacking ranks, though, and have re-opened transfer talks.
St Johnstone boss Tommy Wright accepts every player has his price and an agreement could be reached before the window closes, but has warned Peterborough that they will have to dig a little deeper if they are to secure the signature of a top target.
He said in The Courier: "There has been further communication with Peterborough but no new formal bid. It is not dead in the water and further discussions may take place between the two clubs.
"Stevie is being kept informed every step of the way of their interest, as he has been from the start. There has been no cloak and dagger stuff.
"Three offers have been made but none came close to our valuation of a player we have brought through the youth system and farmed out on loan to progress his career. He is a player I want to hold on to so it would take a very serious offer from any club to lure him away.
"Stevie knows how highly I regard him and that is reflected in the fact he has featured in all five matches this season.
"I wouldn't want to lose him to any club but the chairman does have a valuation on Stevie.
"But right now I have told him he is only a few games into a new season in the top division in Scotland and he just has to keep progressing."