On Saturday, a group of relatives including his wife, son and daughter, daughters- in-law and other members of his extended family met him and discussed his review petition and other family matters, Jailor Nasir Ahmed told bdnews24.com.
Quasem, a Jamaat-e-Islami leader, has been lodged in the Kashimpur Central jail since his arrest in 2012.
He was found guilty by the International Crimes Tribunal and was awarded the death sentence in 2014.
His lawyer said he was almost ready with the draft of the review petition against the verdict.
He said that the review appeal would be made within June 21.
If the review fails, Quasem has the option of seeking presidential clemency, and hangs if that is denied.
According to the law, a review appeal must be made within a fortnight of the publication of the full verdict.
In Quasem's case, the full verdict was passed by the apex Court on June 6, clearing the decks for the Tribunal to issue his death warrant.