Trapattoni wants the 23-year-old to leave Old Trafford in order to get regular first-team football.
Gibson scored his first goal for the Republic of Ireland in the 3-0 victory over Wales on Tuesday night.
“He can do more. He has a big potential. If I was Gibson I would go to another group where I always play,” Trapattoni said on Sky Sports News.
“By doing this you can build your mentality.
“Manchester is Manchester and it is an important club, but sometimes the young players need to play, play and play. It builds confidence.”