The England international has failed to establish himself as the Red Devils' number one and is free to depart Old Trafford before the window slams shut on Friday.
A permanent move to Nottingham Forest was initially considered the most realistic outcome, after he helped Steve Cooper's side stave off relegation to the Championship last season.
However, a breakthrough in negotiations could not be reached between the two sides, and Forest instead turned their attention to ex-Arsenal shot-stopper Matt Turner.
Since the Tricky Trees cooled their interest, Palace - who are likely to lose Vicente Guaita in the coming days - have been pushing to add Henderson to their goalkeeping ranks as competition for Sam Johnstone.
According to The Telegraph, Man United have now given Henderson the green light to undergo a Palace medical on Saturday, having made considerable progress in talks with Roy Hodgson's side.
While a deal is yet to be fully agreed between the two clubs, there is an expectation that Henderson will be a Palace player in the coming days, and personal terms will soon be negotiated.
Barring a brief run in the first team during the 2020-21 season, Henderson has been a perennial fringe player for Man United and has made just 29 senior appearances since joining in 2011.
With Andre Onana immediately taking the gloves off of David de Gea's hands after the Spaniard's departure, Henderson has been an unused substitute in each of Man United's opening three fixtures in the 2023-24 season.
However, the 26-year-old will be expected to challenge for the number one jersey at Selhurst Park, while Man United are already on the brink of signing his successor.
Despite reports of an agreement in principle with Benfica's Odysseas Vlachodimos, the Red Devils are set to complete a cut-price deal for Fenerbahce shot-stopper Altay Bayindir.
Due to a recent back injury for the Turkey international, Man United conducted medical tests earlier this week, and they have intensified their interest after scans showed reassuring results.
Now, Bayindir is believed to be on his way to England to complete his switch to Old Trafford, and Fenerbahce manager Ismail Kartal recently confirmed that the 25-year-old would be leaving the club.
Man United came from two goals down to defeat Nottingham Forest 3-2 on Saturday and are next in action away to fellow top-four hopefuls Arsenal on September 3.