Beasley said: "I can't say where I'm playing, but I do know where I'm going.
"There have been a lot of things said in the papers - rumours this, rumours that. I'm glad I can finally be done with that."
PSV Eindhoven chief executive Stan Valckx told The Sun: "There has been significant progress in the negotiations between ourselves and Rangers.
"The discussions have been accelerated, but we must wait until after DaMarcus plays for his country on Sunday."
Rangers reportedly tabled an initial bid of £500,000 but may have had to move closer to £1m to secure the midfielder's services.
