Burke made his West Ham debut against Queens Park Rangers at the weekend and Allardyce says that he has seen similar qualities in him as he did with Jones.
Allardyce handed Jones his senior debut during his time at Blackburn Rovers and the centre-back has now gone on to become an England international, meaning that Hammers fans will be hoping that Burke can follow a similar path to success.
“When you put a centre-half in, as we did with Phil Jones at Blackburn, you see the quality," Allardyce said. "Can they cope with the pressure as well? Certainly against QPR, Reece did that.”
There have been high hopes about Burke's future at the Boleyn Ground for some time and his initial impression was a good one.
The 18-year-old was not overawed by the occasion against QPR and helped his side collect a point at Loftus Road in a 0-0 draw.
Hammers supporters will be hoping that it is the first of many appearances that Burke makes for the club, as it has been some time since a top-class academy graduate came through.
Allardyce has been criticised for his lack of use of the club's youngsters, but it seems that Burke could be set to be a prominent member of the first-team for years to come.