Elder’s current deal at the ABAX Stadium expires at the start of January, but the Peterborough Evening Telegraph has suggested that the agreement could be prolonged in the new year.
Elder has played a crucial role for Posh ever since joining, and the left-back is enjoying his first taste of League One football.
"I’m loving playing for this football club," he enthused. "It’s a great bunch of lads, it’s a really professional club and it’s really ambitious.
"My loan is up on the 4th January, but it [extending the deal] has not been discussed with me personally. I’m not looking into it too much, and what happens will happen.
"But I want to stay. We have a good platform to kick on in new year, and I want to be a part of that."
Elder has played 19 times for Posh to date, and his place in Graham Westley's first team is under little threat.
The 20-year-old’s impressive form means that he is now an established part of Posh’s starting line-up, and it would be a huge blow for the League One side to lose him in January.
Peterborough have climbed up the standings rapidly since Westley replaced Dave Robertson, and they have won their last six matches.
This sequence of results has increased optimism around the ABAX Stadium, with talk of a promotion push gathering pace.
Without Elder, though, Posh’s options would be short at left-back, as Kgosi Ntlhe is set to be out long-term through injury.