It is understood that Castore has tabled a digital-focused offer, which it hopes will help see off competition from US company New Balance.
Roma are on the hunt for a new kit partner in time for the 2021/22 season, when their revised contract with Nike expires. The club terminated their original deal with the US brand four years early in July.
SportsPro understands the company has offered Roma a digital-focused approach, allowing the club to reach a wider audience and grow their brand internationally. Castore will also look to tap into Roma's rich heritage and US links through its new owners the Friedkin Group, which is led by billionaire Dan Friedkin.
In contrast, New Balance is proposing a more traditional deal, including a lump annual sum plus performance-related bonuses.
SportsPro understands that Roma could make a decision on their new kit partner as early as this week.