The 16-year-old Dutchman, who has signed to race for the Italian squad in 2015, had his first proper F1 test in a two-year old Toro Rosso at Adria International Raceway last week, clocking up enough mileage to be eligible for his F1 Super Licence.
"He finished by doing 396 kilometres without any problem and people were really impressed with his performance," Tost said in Friday's FIA press conference in Singapore.
"Now it’s in the hands of the FIA whether he gets a Super Licence to do the Friday P1 session in Suzuka and then we will see. We will prepare him step by step and I am convinced that he is the correct driver for us for 2015."
Should Verstappen be permitted to drive in first practice at Suzuka, he would be participating in an official Grand Prix session just three days after turning 17.
He will replace Jean-Eric Vergne at Toro Rosso ahead of the 2015 season and will become the youngest driver in F1 history when he makes his race debut in Australia next March.