The 22-year-old former West Ham trainee has scored 23 goals in 97 games for Colchester, but could not save them from League One relegation last term.
Posh director of football Barry Fry has confirmed that his club have verbally agreed personal terms with Moncur.
But newly-promoted Barnsley are also interested as they look to strengthen ahead of a return to the Championship.
Colchester United have accepted bids for midfielder George Moncur from Barnsley and Peterborough United.