Fabio, who spent last season on loan at Queens Park Rangers, made his first start for United since November 2011 in the 2-0 win against Crystal Palace at the weekend.
And the young Brazilian hopes to start more regularly for the Premier League champions, even if it is at the expense of his brother.
Speaking to The Sun, Fabio said: "My brother and I are going to fight for the right-back position.
"Why not? We have always fought over everything, PlayStation, anything. we fight with each other to win. This is no different. Now we are 23 we are mature.
"It's not important whether me or him plays but that we do well. It's a big thing to play for United. We just have to play and enjoy it."