England supporters' group Barmy Army has warned there is "too much risk" in fans travelling to Bangladesh for the team's tour in October.
It says there is "insufficient interest" to merit an organised tour, but will reconsider if fans are given the same security measures as players.
The tour was only given the go-ahead following a security review by the England and Wales Cricket Board.
ECB director of cricket Andrew Strauss says some players may choose not to go.
Moeen Ali, Chris Jordan and Liam Dawson have said they will travel if selected.
Strauss told BBC Sport he had given the players time to think about their decision and would meet with them later this week before the tour party is announced on 16 September.
Limited-overs captain Eoin Morgan is among those who reportedly have doubts, and he said after England's Twenty20 defeat by Pakistan on Wednesday that he had not yet made a decision.
"I have a few more days now to think about it," he told BBC Test Match Special.
"The decision will be based on if I feel safe going out there to play cricket."
Strauss confirmed: "He has had concerns and he has discussed those with me.
"It is an interesting situation with the captain questioning whether to go or not. People view a captain differently because they have different responsibilities. But they are still humans at the end of the day."
England will play three one-day internationals and two Tests starting on 7 October. The matches will take place in Chittagong and Dhaka, where 20 hostages died when a cafe was attacked in July.
Steve Smith, the office manager for the Barmy Army, which organises tours to England away series, said in a statement: "Our view is that supporters should apply the same criteria as the ECB to the tour.
"In other words, unless specific security measures and assurances are put in place, there is currently too much risk attached to travelling to Bangladesh."
Strauss said: "We would hate for anyone to think that because England are going it is safe for anyone to go.
"It is right that people make up their own mind on these things and come to these decisions based on Foreign Office advice, etcetera."
Foreign and Commonwealth Office (FCO) advice is that "there is a heightened threat of further terrorist attacks" in Bangladesh, although a similar threat level applies to many European cities, including London.
The FCO says westerners are in particular danger of being targeted in Bangladesh, adding: "Crowded areas and places where westerners are known to gather may be at higher risk of attack."