Abraham completed a £34m transfer from Chelsea to the Italian capital on Tuesday and has been given the number nine jersey previously worn by Edin Dzeko, who has joined Inter Milan.
Abraham came through the youth ranks at Chelsea when Mourinho was manager at Stamford Bridge and has previously spent loan spells at Swansea City, Bristol City and Aston Villa.
Having lost his place in the Chelsea set up under Thomas Tuchel and subsequently in the England squad for EURO 2020, Abraham has been backed by Mourinho to show his true mettle after making the move to Italy.
"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. I know how he made the decision to leave the Premier League, which is always tough for an English player.
"That tells me so much about him, because when you leave the Premier League, you do so because you're ambitious.
"You leave because you want to get back in your national team, because you want to play at the World Cup, because you want success outside of England, where not many English players have had brilliant careers.
"He comes here with that ambition and we hope to see his best qualities as a player."
Roma will start their Serie A campaign at home to Fiorentina on August 22.
