However, Saints may have to reach a financial agreement with Bolton over the free agent.
The 20-year-old rejected a contract extension offer from Bolton in the summer after being frustrated with his lack of first-team action.
Saints manager Danny Lennon told the club's website: "We really like Aaron Mooy and we have made him an offer which he has accepted.
"Bolton still hold rights of compensation over him and there could possibly be complications that are outwith our control but we'll be having discussions with them."
Lennon added: "We won't be following up our interest in Manny Giannaros so he is going back to Australia."
Source: Team Talk
Source: Team Talk