Allardyce, despite his own uncertain future at Everton, is looking to Jones to reinforce his defence for next season.
Allardyce and Jones have a long history, and the United defender was even given his professional debut in 2009 at Blackburn Rovers by the Everton manager.
However, having been plagued by hamstring injuries this season, Jones has not yet appeared in the starting squad for Mourinho's side since last month's defeat at the hands of Newcastle United.
Allardyce is looking to test whether Jose Mourinho will want to hold onto the 26-year-old in the summer transfer window, that is, if the Everton boss is able to hold onto his own job.
For his part, Jones may well be tempted with a move to regular first-team matches given that United have no shortage of players vying for the defence positions, including Eric Bailly, Chris Smalling, Marcos Rojo, and Victor Lindelof.
Everton will certainly need to strengthen their defence having conceded 49 goals from 30 matches this season but will have to buy intelligently as they look for the most viable investment for their club.