The two fighters will go toe-to-toe at Wembley Arena on Saturday evening after number one seed Groves chose the fellow Brit as his first round opponent.
However, coach Costello, who has been in Cox’s corner for 11 years, believes the southpaw-fighter has picked the wrong man.
“We always wanted this fight with George Groves,” he said.
“We’d asked for this fight before so when George picked Jamie in Monte Carlo we was in our element.
“All our Christmases had come at once.
“I’ve said it before and people have laughed at me. I’ve believe Jamie can beat any 12 stone fighter on the planet.”
Groves is expected to weigh in heavier than Cox, but Costello does not believe that it would make any advantage in the ring.
“People are talking about size but that has no relevance for me,” Costello added.
“Carl Froch was 12 stone four when he fought George Groves at Wembley, and I believe George was over 13 stone.
“Jamie will be heavier than 12 stone four. Jamie would be dehydrated if he got into the ring at 12 stone four.
“Size had no relevance that evening and it will certainly have no relevance on Saturday night.
“Jamie’s ability and what he brings will be very relevant because we’re taking that WBC Super Championship home with us.
“That’s 100% set in stone. We’re here to a do a job and that’s what we are going to do. George Groves made a big mistake when he picked Jamie Cox.”