Both Hull players at the time, Maguire and Robertson have gone on to enjoy whirlwind years - the centre-back integral to the England side that finished fourth at the 2018 World Cup and the left-back impressing as Liverpool reached the Champions League final in May.
Whereas Everton failed to properly address their problems in the pair's positions, and spent the bulk of this summer's transfer window trying to put that right - finally landing Yerry Mina and Lucas Digne from Barcelona.
According to Allardyce, who was sacked by Everton in May after six months in charge at Goodison Park, they could have quite easily saved themselves the hassle.
"I was told by Steve Walsh who was the director of football when I got there that (there) was a very big opportunity for them to take Maguire and he said Robertson, both from Hull City. What a pair they would have been at the time," Allardyce told the Alan Brazil Sports Breakfast show.
"You can ill afford any slip-ups (when it comes to recruitment) - they would have made a big difference."
Discussing Maguire in particularly shortly afterwards, Allardyce laid the blame at the feet of his predecessor.
"I think Ronald (Koeman) didn't quite fancy it at the time," added the former Everton manager. He could have been a big asset. I don't think anybody saw Harry's progression (happening) as quick as it did. (He's) one of the top centre-halves in Europe I'd say."
Will Mina and Digne prove better signings for Everton than the Leicester City and Liverpool pair?
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