It is believed that a fee of €6m (£5.1m) has been agreed for the Dutch international, who has been linked with a move to Rome for several weeks.
The 28-year-old is due to undergo a medical at Roma this weekend and will then begin discussions over personal terms.
"It is finally time to discuss the contract of Maarten," Stekelenburg's agent, Rob Jansen, told Sport Promotion. "This has all taken way too long."
Stekelenburg has only played for Ajax during his club career but he has 40 international caps for the Netherlands.