According to NDTV, Kausikan recounted the episode while responding to a Pakistani journalist during an interaction with journalists at the National Press Foundation.
The former diplomat said the hijackers wanted to speak with Bhutto, who was out of office at the time.
Kausikan said he eventually reached her residence by telephone but was told by a household staff member that "Madam is sleeping and cannot be disturbed."
He said he ended the call immediately, adding that the incident left a lasting impression on him.
During the discussion, Kausikan also rejected the argument that Pakistan's challenges stem primarily from its geographical location between India and Afghanistan.
"You cannot blame location on everything. That's an excuse," Kausikan was quoted as saying by NDTV.
"Pakistan has been mismanaged terribly since the beginning. Its politicians are a waste of time, all of them, regardless of parties, and the military is a big part of the problem," he said.
He also described Pakistan as "a state teetering on the brink of failure", arguing that recent diplomatic gains would not address the country's structural economic and governance challenges.
According to NDTV, Kausikan said Pakistan's recent diplomatic successes may have improved its international standing but would do little to resolve its internal crises.
Following failed negotiations during the 1991 hijacking, Singapore commandos stormed the aircraft, killing all four hijackers and rescuing the passengers and crew, NDTV reported.