Garcia revealed Bradley's exit after watching his side progress to the quarter-final of the Coppa Italia with a 1-0 win over Sampdoria.
The French head coach gave a debut to new signing Radja Nainggolan in the game and his signing has paved the way for Bradley's departure.
Bradley has found first-team starts hard to come by this season, prompting the United States international to request a move from the Italian capital.
"It's a shame Bradley is leaving," Garcia said. "But he wanted to go and we cannot keep hold of a player who doesn't want to be here.
"It's a pity and I understand he wanted more playing time, as it's difficult to get space in our midfield.
"We wish him all the best in his new adventure."
Garcia was pleased with the performance of his team against Sampdoria and the impact of Nainggolan.
"I liked the performance, as we had plenty of chances to double our lead and did not suffer at the back," he added.
"It’s a shame the second goal didn't arrive so we could be more relaxed at the end, but I am happy with our qualification, as that was the important thing.
"Nainggolan can play in any midfield role. With him there we can play with Miralem Pjanic in a more advanced position and Francesco Totti as trequartista. There are many possibilities with Radja."
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