Doni joined the Reds from Roma in 2011 but made just four first-team appearances, his last involvement coming on the last day of last season at Swansea.
The 33-year-old Brazilian did not feature in the current campaign and slipped further out of the picture when Brad Jones was favoured as a replacement for the injured Jose Reina.
Liverpool also allowed Daniel Pacheco to leave the club on transfer deadline day, the 22-year-old heading to Spanish second division club SD Huesca on loan until the end of the season.
