“I have sought to be as open and honest as I can be about this matter,” he said.
“I have told you everything I know about this matter.”
The text messages were between Ms Higgins and another Liberal Party staffer in early April of 2019, soon after the alleged rape took place.
In the texts, the male staffer wrote that he spoke to the Prime Minister’s Office (PMO).
“He was mortified to hear about it and how things have been handled,” the man’s text message said.
“He’s going to discuss with COS (chief-of-staff) – no-one else.”
It is not clear who the “he” is in the text messages.
The prime minister said the reports from this morning would be part of an internal review into the handling of the alleged rape.
“I set out in the parliament this week the timetable of when I am advised my office knew about it,” he said.
“I knew about it on Monday.
“It shattered me. It absolutely shattered me.”
Mr Morrison said he has been listening to the issues raised over the alleged sexual assault, and is seeking to “address them in the best possible way I can”.
“We have to deal with issues of culture which I’m sure you would agree are not confined to the offices of parliamentarians,” he said.
“This is an issue that every workplace deals with.
“I have no doubt it occurs in the media workplaces, offices in law professions or whatever it might be.”
Opposition Leader Anthony Albanese said Mr Morrison’s explanation “doesn’t pass the pub test”.
“It’s not credible that the prime minister continues to say that his office only found out about it this week,” Mr Albanese said.
“And the dissembling nature of his answers before the Parliament are not good enough.”
Mr Albanese said Ms Higgins deserves better than the way she was treated by the government this week.
“For her to not get the respect of getting straight and clear answers about what the response was at the time, about who was told and what they were told at the time, is disrespectful of Ms Higgins,” he said.
“I believe Brittany Higgins.”
Ms Higgins alleges that she was taken into the office of then-Defence Industry Minister and current Defence Minister Linda Reynolds by a government staffer when she was drunk and then raped.
Days later Ms Higgins was called into a meeting with Senator Reynolds to discuss the alleged rape in the same room.
Ms Higgins is considering pressing charges against the alleged rapist.
The alleged rape took place little more than a month before the 2019 federal election.