Torreira, who arrived from Sampdoria in 2018, is set to link up with Diego Simeone’s side.
He does not feature in Arsenal manager Mikel Arteta’s long-term plans at the Emirates and there was hope that he would be sold outright this summer.
However, a loan deal has instead been struck, with an option to buy inserted into the deal.
Arteta has been told to trim his squad as he looks to add new recruits in the latter stages of the transfer window and Torreira has been shown the door.
He bid farewell to those at Arsenal’s Colney training base on Friday and will travel to Madrid in the next 24 hours, according to Goal.
Torreira’s exit could pave the way for new midfield arrivals at the Emirates.
Chelsea star Jorginho has been linked with a loan move, while Arteta is still determined to land Lyon midfielder Houssem Aouar.
Sead Kolasinac, William Saliba and Sokratis Papastathopoulos are among those who could be shipped out.
‘We need to make some decisions in the next three days, I think everything has been delayed so much, not just us but probably every team because of the context that we’re in at the moment.
‘We will try to make the best possible decisions and at the end, whatever we have, we go for it fully convinced and prepared to have a good season.’