Morgan, Test captain Alastair Cook and team members will meet the England and Wales Cricket Board's security director Reg Dickason on Thursday to discuss the matter.
An attack in Dhaka last month claimed by the Islamic State group saw 20 hostages killed, including nine Italians, and Australia have cancelled senior and under-19 tours of Bangladesh in the past year.
"They'll debrief us on the findings on both tours (Bangladesh and India) and we'll chat about it. It has to be quite open," Morgan said in comments reported by British media on Wednesday.
"There's always room for a personal decision. You also have to understand guys might have different priorities."
England will play three ODIs and two Tests in Bangladesh in October and November before heading to India.
Bowlers Andy Caddick and Robert Croft pulled out of England's tour of India in 2001 for security reasons, while England forfeited a World Cup match against Zimbabwe in 2003 after refusing to play in Harare.
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