The current West Ham United boss oversaw part of Jones' development at Ewood Park so knows him very well and feels that he has what it takes to make it at Old Trafford.
"His mental strength and the abilities to cope with the pressure of the Premier League at such a young age stood out," Allardyce told BBC Sport.
"He wasn't fazed, didn't look too nervous and was quite calm.
"That is very important going into the biggest league in the world at such a young age.
"I think he made his (Premier League) debut against Chelsea and was man of the match.
"His mental strength was always going to be good enough and that is one of the biggest factors if you are going to be a top, top player.
"I think eventually, with a little more experience, which is going to gain more quickly at Manchester United, he will ultimately become an England defender."