The England international, according to Sky Sports News, has passed a medical ahead of a £20m switch to Goodison Park.
Walcott has been an Arsenal player for 12 years, and has scored more than 100 goals for the North London outfit during that time.
However, in recent seasons the 28-year-old has had to settle for a bit-part role under manager Arsene Wenger, who signed the winger from Southampton in 2006.
While addressing the media in a press conference, Tosun said: "Theo is a great player, proven player. He's done really well for Arsenal.
"He's really fast, he's a great crosser. I would say it's a massive signing for us and I think I'd link up well with him. I'm sure if we sign him he would have a great input."
Tosun signed for Everton on January 5 in a reported £27m deal from Besiktas.