Argentine left-back Silva is set to make the move permanently for next term, in a deal Sporting revealed could be worth €5.7 million.
It leaves the Giallorossi well in profit, having received an initial €20m for Emerson.
"I am very happy to have joined Roma, for me it's an incredible feeling," Silva said.
"I am very motivated: I want to make my contribution among this great group of players."
Silva has made 12 appearances in all competitions for Sporting this season, including four matches in the Champions League.