Ajose, 20, has been on trial at Boro for the past week and featured in the first half of the League One club's 2-0 pre-season friendly defeat to Tottenham on Wednesday evening.
Speaking after that game, Stevenage boss Gary Smith said: "Nicky has come in for us to have a look at him from Peterborough. He's still under contract at Peterborough.
"There may be something we can do there in terms of a deal, but that'll be going on behind the scenes with the chairman."
Manchester United youth product Ajose is currently on the transfer list at Peterborough.