Peacock-Farrell has become an ever-present in the first-team squad this season after Ross Turnbull broke his ankle in November.
Steve Evans has primarily used the 19-year-old as understudy to first-choice Marco Silvestri but handed him his senior debut against QPR this month.
The Yorkshire Post claim United are keen to tie him down long-term after he signed a 12-month deal with the club last summer.
Evans has since suggested there will be more opportunities for the teenager in the first team as the season comes to a close.
“He’ll hopefully get involved to start a game before the end of the season,” he said.
The youngster is said to be considering the two-year deal but could leave the club if first-team football isn't forthcoming.
Current No.1 Silvestri still has two years left to run on his deal and is firmly established as Evans' first choice
And former Chelsea man Turnbull is due back from his ankle injury in the summer and still has 12 months on his contract.