The 19-year-old spent the majority of last season on loan at Bury - scoring 13 goals in 28 league games to help the Shakers seal automatic promotion to League 1 - and has penned a four-year contract with Darren Ferguson's side.
He becomes the Posh's second arrival from his father's outfit in less than a week, following the recruitment of midfielder Ryan Tunnicliffe on a six-month loan deal.
While United defender Scott Wootton has also joined Peterborough's training camp in Portugal with a view to signing later this month.
"Nicky has good pace, is two-footed and a very good finisher. He has a lot of potential, has good pedigree and was successful during his loan spell with Bury last season, scoring a lot of goals in League 2," Darren Ferguson told Peterborough's official website.
"We feel he can make the step up to the Championship. He has a fantastic attitude, speaking to my Dad, he tells me that he is the first in the building every day so that stands him in good stead.
"I am looking forward to working with him. Obviously Craig Mackail-Smith has left [to join Brighton] and I am not going to be putting any pressure on the lad to be the replacement or anything like that, but he has that pace, he plays on the shoulder and he can also link play. He can play wide if we went to a three up top and I believe he is very good signing for the club."
Ajose admitted that he had no hesitation in agreeing a move to the newly-promoted Championship outfit, and said he would not be putting any pressure on himself to replace the goals of Mackail-Smith, who bagged 80 league goals in four-and-a-half years with the club.
“I have heard and seen what Craig Mackail-Smith did for Peterborough last season, scoring the amount of goals he did was unbelievable," he said in the Peterborough Evening Telegraph.
“I wouldn’t say I was a replacement for him, I just want to come and do my bit for the team. It is a massive move for me. To be playing Championship football at 19 was a big attraction. Manchester United is a great club, but I have been there for 10 years, so now it is time to get into the real world.
“Once you have had a taste of first team football as I did at Bury, you don’t want to go back to playing in the reserves. When I got offered the opportunity to come to Peterborough it was a no-brainer really.”
And he was also quick to pay tribute to Bury, without who he would not have earned his move to London Road so quickly.
"I'd like to say a massive thank you to all Bury fans for their support last season, and especially for the way that you all welcomed me to Gigg Lane," he told Bury's official website.
"Also, a massive thank you to Alan Knill and Richie Barker for giving me the chance in league football. My time at Bury is something special to me and something that I will always cherish.
"I am very grateful for the chance and you can be sure that the first result I look for in the future is Bury's. I hope that Bury have another great year."
NEWS:Ajose set to sign for Championship side
