Abraham has signed a five-year contract with the Serie A club to fill the void left by Edin Dzeko's departure to Inter.
Roma paid €40million (£34.1m) to secure the services of a player who made his Chelsea debut in 2016 and has played six times for his country.
Mourinho believes leaving the Premier League behind for a new challenge shows Abraham has lofty aspirations.
"With Tammy, I prefer to say, 'wait and see'," Mourinho said in a video posted on Roma's Twitter account.
"I say that with total confidence. I've known him since he was a boy. He's never played for me because he was a 14, 15, 16-year-old boy when I was at Chelsea, but I know him very well. I know him as a player, a person and in terms of his mentality."