The 29-year-old, who was made available for transfer by Preston in the summer, has not played for the club this season.
Rovers boss John Ward has said that the arrival of Beardsley will boost his striking options.
"He is an experienced player and he will give us what we do not have at the moment, and that is experience down the middle of the pitch," Ward told the club's official website.
"He provides that extra bit of knowledge. He is a strong boy and he is a team player who works very hard for his team.
"He may not get the headlines for scoring all the goals, but he brings people in to the game and creates chances. He should be a big help to the young lads in the squad."
Beardsley scored three goals in 22 appearances for Preston last season.