None of those horses are considered among the pre-race favourites, but the training duo said that is exactly the way they like it.
"For me the horses to beat are So You Think and Game On Dude. These are the best two horses in the race," Al Zarooni said.
He sends Capponi, the Al Maktoum Challenge Round III winner, and Monterosso, the Dubai City of Gold victor, to the post for the US$10 million (Dh36.7m) race. Bin Suroor saddles Prince Bishop and Mendip. Al Zarooni considers it a bonus that his horses are not attracting as much attention as Aiden O'Brien's charge So You Think or the American raider Game On Dude.
"We are outsiders and I prefer outsider - no pressure," he said yesterday. "When you are the favourite it is very difficult. You have to win with the favourite, but if you have the outsider and you lose then you've lost nothing."
And Al Zarooni said he does not think that his horses are without a chance of claiming Godolphin's sixth Dubai World Cup, a race that the Dubai racing operation has not won since 2006.
"Obviously I think they have a chance otherwise they would not be in the race," he said. "That's why we are here."
Capponi is to be ridden by Ahmed Ajtebi, Godolphin's Emirati jockey.
"Capponi is improving after each race but the way he won the Maktoum Challenge for me was a big improvement," said Al Zarooni, whose charge beat Doug Watson's World Cup contender Silver Pond on that occasion.
"We are very happy with him and he's a front-runner which is why I put Ahmed on him because I think he is good in front.
"I think Mikail Barzalona will sit behind a bit more and wait so to have them the other way round would be wrong."
Barzalona partners Monterosso, who ran third behind the Japanese horses Victoire Pisa and Transcend in last year's Dubai World Cup.
"We ran him once this season and he needed the race and Mikail didn't push him," Al Zarooni said. "He finished fourth and we brought him home and afterwards we saw that he was moving much better for it. I don't know if he's a better horse than he was last year but I can say that he is at the same level."
Bin Suroor said both his World Cup runners were heading in the right direction.
"Prince Bishop won a big handicap on his first time out. He was third in the third round of the Maktoum Challenge. He's an improving horse and Frankie Dettori is pleased with how he's going."
Mendip is also expected to put up a good showing under Silvestre de Sousa.
"They are both good horses," bin Suroor said. "Mendip won the Al Maktoum Challenge Round II but in his last run he got quite upset and was kicking the gate but when we got him home he was much better and we are pleased with his progress."
Bin Suroor, who has won a record five Dubai World Cups and was the most successful trainer in the Carnival this season, said it was not an easy race to win.
"Everybody knows that in the past Godolphin won five times but this is the hardest and most difficult race to win. It has the best horses in the world but we have four horses in the race and we hope that we can win," he said.
Bin Suroor said his best chance of victory on the World Cup card will come from African Story, who will be ridden by Dettori in the Godolphin Mile. Al Zarooni felt most confident about Opinion Poll, who also goes under Dettori, in the Dubai Gold Cup.
"This is a race that we can win," said Al Zarooni of the horse that finished second to bin Suroor's Gold Cup rival, Fox Hunt, in the Group 3 Nad Al Sheba Trophy. "Opinion Poll needed the run, and has come on for it. He's fitter now, and I think he represents our best chance."
Follow us @SprtNationalUAE