Allardyce introduced Jones to first team football at Blackburn Rovers.
Ahead of United's clash with Everton, Jones told Sky Sports: "He [Allardyce] was the one who gave me the chance in the Premier League and gave me my debut and wasn't afraid to put me in, albeit there was no one left.
"I owe him a lot of thanks for that. I will be forever grateful.
"As a young player, you need as much support as possible and he gave me that and gave me the chance."
He added: "People might not like the style of football he plays and may have moans and groans about it, but ultimately he gets results."