The 19-year-old has been given until Sunday to accept or reject an extension to a deal which runs out at the end of June.
Leeds opened negotiations with Peacock-Farrell midway through this season and the club want to tie him down to the summer of 2018 after seeing him break into their first-team squad.
Head coach Steve Evans named him on the bench throughout the second half of the term and gave him a full debut in a 1-1 draw with Queens Park Rangers last month.
Despite promising to use him again before the end of the campaign, however, Evans stuck with first-choice Marco Silvestri during Leeds’ last few Championship games.
Peacock-Farrell has been considering United’s contract offer for a number of weeks but he is understood to be seeking assurances about the club’s plans, both for him and the coaching team at Thorp Arch.
Current goalkeeping coach Richard Hartis, who has worked closely with Peacock-Farrell in the past 12 months, is leaving Elland Road to take up a youth-team position with England and a replacement for him is yet to be announced.
Evans could also move on in the coming weeks with United owner Massimo Cellino showing no sign of renewing the Scot’s short-term deal.
Peacock-Farrell’s situation is the most pressing among the senior squad after Mirco Antenucci - one of only three first-team players who are out of contract next month - confirmed over the weekend that he would be leaving Elland Road.
United are due to release a retained list of development-squad players shortly and the club are expected to offer new deals to the likes of Tyler Denton, Frank Mulhern and Alex Purver.
The YEP understands, however, that Lewis Walters - a forward who was on the brink of his debut in 2014 - is likely to be amongst the list of youngsters moving on.