The future of the tour was in the balance after an attack on a Dhaka cafe in July saw 20 — mostly foreign — hostages killed, with the Islamic State group claiming responsibility.
However, the England and Wales Cricket Board (ECB) said last month that the tour, which includes three one-day international and two Test match fixtures in October and November, would go ahead as planned.
The confirmation followed a visit to Bangladesh by an ECB delegation led by their long-serving security chief Reg Dickason.
However, no England player has yet publicly confirmed their willingness to tour Bangladesh.
Following a security briefing given to players by Dickason on his return, several British press reports suggested Morgan had been reluctant to commit himself to the tour.
However, Morgan — in common with several members of England’s one-day squad — has said his attention for the time being is focused on the ongoing series with Pakistan.
England’s four-wicket win in Leeds on Thursday left them 4-0 up in a five-match series ahead of Sunday’s finale in Cardiff.
Morgan is also due to captain England in a lone Twenty20 international against Pakistan at Manchester’s Old Trafford on Wednesday.
ECB director Andrew Strauss, the former England captain, has been at pains to indicate players won’t be rushed into making a decision on whether to go to Bangladesh, although Test skipper Alastair Cook has reportedly already made himself available.
But with the team due to fly to Dhaka for the one-day leg of the tour on September 29, Morgan accepts the time for a decision will probably have to come soon after Wednesday’s match, England’s last international of their home season.
“I would certainly like to think probably within a couple of days after that there will be something definitive,” Morgan told reporters at Headingley.
“I have not made my mind up. I will be taking as much time as I need.”
Asked about his willingness to tour, Morgan said: “Reluctant isn’t the word.
“I think when so much information is thrown at you within two months of a terrorist attack you need to digest everything and be settled within yourself so you can go there and concentrate on cricket.”
The Irishman added: “We are in the middle of a series. We have not had a great deal of time to think about it.”
Morgan insisted he did not expect other players in the side to follow his lead on Bangladesh because of his position as captain.
“Absolutely not,” he said. “It is a completely individual thing.
“The decision any individual makes within the side is the right one, and everybody will stand by whoever wants to go or not. I think that is very important.”
Meanwhile Morgan is presiding over an impressive transformation in England’s limited overs form after their embarrassing first-round exit at last year’s 50-over World Cup.
Victories in Cardiff and Manchester would see England complete a first unbeaten home season in white-ball cricket and Morgan said: “That will be a huge achievement.
“The performances we have put in, particularly (in Leeds) when we were not at our best, have shown great character and depth within the side.
“I think our best is good enough to beat most sides.
“It is just when we are off it is important other guys fall in behind — and (this win) proved that.”
England slumped to 72 for four at Headingley chasing 248 before fifties from Ben Stokes and Jonny Bairstow got them back on track.
“If we were chasing 290 it might have been a different story,” said Morgan.
“We might have had to press the button a little bit earlier but certainly Ben Stokes’s innings showed a lot of maturity. Ben is a hugely talented player but very aggressive too.”