The tour, which begins on September 30 and will include two Tests and three ODIs, had been called into question after an attack in July on a cafe in Dhaka in which 20 hostages were killed, including 18 foreigners.
Morgan had already spelled out the reasons for his personal concerns about travelling following his previous experience of security alerts in both India and Bangladesh.
And the ECB confirmed late Sunday that both Morgan, 30, and Hales, 27, had let England cricket chief Andrew Strauss know their decision this weekend.
"We have had open and honest conversations with all the players about the security arrangements in Bangladesh and at this stage we are not expecting any other individuals to withdraw from the tour.
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"Final selection for the Bangladesh tour, will now take place on Friday."
Strauss added: "As with all England overseas tours, the safety and security of players and staff is of the utmost importance to ECB.
Lancashire wicketkeeper-batsman Jos Buttler will captain
Buttler, the vice-captain, has led England only once before in a Twenty20 international against Pakistan last winter.
Following the July attacks the ECB sent an inspection team to Bangladesh led by their long-serving security chief Reg Dickason.
After he reported back, the board confirmed last month that the tour, which is scheduled to run until November 1, would go ahead as planned.
"A thorough and detailed risk assessment was approved and accepted by the ECB Board and the England players and management have been fully briefed on all aspects of the safety and security arrangements," the ECB statement added.
"In 2010 we played an IPL game in Bangalore and a bomb went off in the ground. We immediately left for the airport. So that was an instance. Another one was (in) Bangladesh playing domestic cricket during political elections when things were incredibly violent."
The Irishman added: "I think ultimately, as I've said before, as an individual you need to be comfortable within yourself to focus on cricket.
Test captain Alastair Cook is understood to have given a private assurance he will be on the tour, while coach Trevor Bayliss, who was on the Sri Lanka bus that came under armed attack in Lahore in 2009, has stated publicly he will travel to Bangladesh.