Haroon Lorgat would be present during the series to show solidarity and support for the people of India and the two teams.
"Safety and security of the players, officials and spectators must be the key consideration at all times," said Morgan.
"But if security assessments indicate that the measures put in place are appropriate then we - players, officials and all stakeholders - have a duty to ensure cricket takes place.
"I have been in regular contact with both the President of the Board of Control for Cricket in India (BCCI) Shashank Manohar and Giles Clarke, the England and Wales Cricket Board's (ECB) Chairman, and it is clear they both want to play the matches if at all possible.
"The players have also demonstrated an acceptance of their role in trying to put a smile back on the face of a country that has undergone a horrendous experience.
"ICC Chief Executive Haroon Lorgat will be in Chennai for the first Test and I will be in Mohali to show the ICC's support for India and the two teams.
"I would urge as many supporters as possible to join us during the series, not only because they will see two excellent sides playing Test cricket, the pinnacle of our sport, but also because by doing so we will demonstrate that we will not be dictated to by terrorists.
"I believe the Test series will be a clear demonstration of the resilience of the Indian people and the game of cricket and an illustration that the game has the ability to bring people together.
"As Haroon said earlier this week, cricket is a common thread that binds all ICC Members and we must seek to ensure those ties endure. By doing so we will ensure our strong sport will grow even stronger," added Morgan.