It is understood Everton will pay £1.5million up front for the 18-year-old Bowler, who made his Rangers debut in May, with a further £2.5million in add-ons.
QPR are reluctantly selling Bowler, who has only played 21 minutes of senior football, after rejecting an initial £500,000 offer from Everton for the highly-rated winger.
It is believed Bowler is having a medical with Everton and Rangers director of football Les Ferdinand claimed the Championship club tried hard to persuade him to stay.
“I had a chat with Josh and his dad in May and told him that he would play for our first team quicker than Everton’s,” Ferdinand told Standard Sport.
“I made all the points about the advantages of being here at QPR and hoped it resonated with him.
“We hoped to keep hold of him, but when a Premier League club comes knocking on the door, they have the finances to make a move.”