Arsenal could wrap up their entire summer signing campaign by the weekend.
The Spaniard has already secured the signing of former Juventus full-back Stephan Lichtsteiner on a free transfer, but is now looking to strengthen the team further.
The London Evening Standard says the three transfers would use up all of Emery's £50million budget.
Sampdoria midfielder Torreira is likely to cost £26m while German shot-stopper Leno will be no cheaper than £20m. Borussia Dortmund defender Papastathopoulos is expected to join for around £16million.
If Emery wants to bring in more transfers then it is thought he will have to fund them by selling some of Arsenal's stars.