Twenty hostages and two police officers died during a 12-hour siege when Islamic militants stormed a cafe in the capital city Dhaka on Friday.
England are scheduled to play three one-day internationals and two Tests in Bangladesh.
The ECB said that "safety and security of players was always paramount".
England will play all their fixtures in Dhaka or Chittagong during their month-long visit, which begins with an ODI on 7 October.
The ECB said it will consult with the Foreign and Commonwealth Office over security in Bangladesh, and would take "appropriate steps" if advised that "the environment is not acceptably safe".
The England and Wales Cricket Board says it is monitoring the security situation in Bangladesh before England's tour in October.