The 24-year-old joined Stanley on an emergency loan deal at the end of last month to solve their goalkeeping crisis.
Regular number one Ian Dunbavin is out for the season with a shoulder injury and his initial replacement, Paul Rachubka, returned to Leeds after a red card against Plymouth and subsequent three-match ban scuppered his own loan spell.
Belford's deal expires before the Reds' trip to Aldershot this weekend and, with Rachubka's suspension now served, manager Leam Richardson would appear to face a straight choice over which keeper he commits to for the remainder of the campaign.
"I've enjoyed coming here, they're a good set of lads," Belford told the Lancashire Telegraph.
"I probably would like to stay if that's possible because I'm playing football. I'd rather be here than sitting on the bench at Bury."
He added: "I'm out of contract at the end of the season and I've not really been in the side there. I spent the first month on loan at Southend this season and spent the last few months at Southend last season. I've been out of the picture for a while now.
"It's up to the gaffer and Bury, they've got to sort it out themselves but I would like to stay here as long as possible really."