The Reds have offered Tony Gallimore a contract until the end of the season.
The 31-year-old is recovering from a knee operation which has ruled him out since December 13 but is back training and is expected to be back in the first team this month.
Burns, Handyside and Ireland have already signed.
Barnsley are also currently trying to agree a deal with Manchester United to extend striker Daniel Nardiello's stay beyond a month.