The former chief press secretary to Margaret Thatcher wrote the comment in a letter to Liverpool fan Graham Skinner in 1996 after the broadcast of Jimmy McGovern's Hillsborough.
In the correspondence, which was handed to the Hillsborough Family Support Group (HFSG) this week, Ingham said: "I have, however, one suggestion to make: for its own good, Liverpool - with the Heysel disaster in the background - should shut up about Hillsborough.
"Nothing can now bring back those who died - innocent people who, by virtue of being in the ground early, had their lives crushed out of them by a mob surging in late."
Ingham was asked by the Liverpool Echo if he thought he should apologise in light of the Hillsborough Independent Panel report again confirming that the deaths at the stadium were not caused by the actions of fans.
Ingham replied: "What have I to apologise for?"
He added that he had not read the panel's findings and said that he had formed his own opinion of it from media reporting following its publication.
Ingham also claimed that political games were being played.
HFSG chair Margaret Aspinall said of Ingham: "Just like the woman, he is not for turning.
"We know the truth and deep down he knows the truth, he just can't face it. We have proved that he's wrong."
Former editor of The Sun Kelvin MacKenzie apologised for his infamous 'The Truth' headline following the publication of the Hillsborough Independent Panel report.
However, he later demanded an apology and compensation from the South Yorkshire Police for the "lies their officers told", which was refused.
The Sun also apologised for its role in spreading lies about the stadium disaster under the headline 'The Real Truth'.
Current editor Dominic Mohan said: "Twenty-three years ago The Sun newspaper made a terrible mistake. We published an inaccurate and offensive story about the events at Hillsborough.
"We said it was the truth - it wasn't. The Hillsborough Independent Panel has now established what really happened that day.
"It's an appalling story and at the heart of it are the police's attempts to smear Liverpool fans."