The frontman, who has been strongly linked with Tottenham in the past, has struggled for fitness and form since returning from the World Cup, with new Sevilla coach Gregorio Manzano preferring Fredi Kanoute and Alvaro Negredo in attack.
Fabiano has only started once and has made three substitute appearances in La Liga this season and has yet to score a goal.
The 29-year-old striker signed a new deal at the club in August after being linked with a move away from Sevilla.
However, Fabiano has revealed he could move on if his situation does not improve in the New Year - news which is sure to alert a number of interested clubs.
"I'm going through a tough period, but it isn't the end of the world," Fabiano told the club's official website.
"I'm not thinking about leaving in January. I just want to play again. January is a long way away. I don't know what will happen. If things remained the same in January I would seek a solution.
"I want continuity. I don't want to play one game and spend the next on the bench.
"I don't understand something that isn't good for Sevilla. It isn't good for me either."
Meanwhile, Fabiano's agent, Jose Fuentes, has denied rumours that he has been in talks with interested suitors about a move for the former Porto hit-man.
"I have never said that I was having negotiations for making Luis Fabiano leave Sevilla," Fuentes told Punto Radio.
"It's not true that we are thinking in the winter market and nothing like that.
"He has renewed his contract until 2013 so it's stupid to think at this point that he is going to be transferred.
"These rumours make people think that we don't know how to represent our players and it's a negative rumour for Luis Fabiano.
"I am convinced that Luis is going to be a regular player for his team again.
"It's normal that he doesn't like to be on the bench, no player likes to be a substitute, they all like to play.
"He has to make an effort for not being on the bench, it depends on him.
"He doesn't have any problem with anyone. The coach has taken his decision and Luis accepts it, besides that maybe he doesn't agree with that decision."