Unai Emery's club would have to pay £26.4 million for the Uruguayan, and if they do, it could work out as a double boost for rivals West Ham.
Here is a look at the two ways the Hammers could benefit from Arsenal signing Torreira.
Arsenal ace Jack Wilshere has been strongly linked with joining West Ham and Torreira's possible arrival pushes him closer to the exit.
Wilshere's contract expires at the end of this month but there are no signs of progress in terms of renewing his contract.
Arsenal choosing to negotiate with Torreira as a priority tells it's own story.
Jean Michael Seri
Before pushing for Torreira, Arsenal had been reportedly eyeing up Nice midfielder Jean Michael Seri, SkySports reported.
West Ham are also claimed to be keen on Seri, with The Sun reporting Manuel Pellegrini is keen on a move.
Chelsea are also lurking, but with their managerial situation unclear and Arsenal now expected to drop out of the race, West Ham have an opportunity to strike now and try and bring in the impressive midfielder.