The Mail on Sunday reported private messages between Ms Truss and foreign officials, including about the Ukraine war, fell into foreign hands.
The hack was discovered during the summer Tory leadership campaign but the news was suppressed, the paper said.
The government said it had "robust" cyber-threat protection in place.
The spokesperson added that the government "did not comment on individuals' security arrangements".
Levelling Up Secretary Michael Gove told Sky News he did not know the full details "of what security breach, if any, took place" but said the government took these issues "incredibly seriously".
Details about the hack were suppressed by then-prime minister Boris Johnson and Cabinet Secretary Simon Case, The Mail on Sunday claimed, citing what it said amounted to a "news blackout" imposed by Mr Case.
The newspaper also said private messages exchanged between Ms Truss and Kwasi Kwarteng, her close friend whom she made chancellor when she became prime minister, were also uncovered by the alleged hack.
It is not clear how any hack happened, but opposition parties have seized on the issue.
"There are immensely important national security issues raised by an attack like this by a hostile state which will have been taken extremely seriously by our intelligence and security agencies," said shadow home secretary Yvette Cooper.
"There are also serious security questions around why and how this information has been leaked or released right now which must also be urgently investigated."
The Mail on Sunday reported agents suspected of working for Russia had been responsible for the alleged hacking, citing unnamed sources, but the BBC has not been able to verify this.
The Mail on Sunday claims private messages were accessed in the hack (archive photo)
The Liberal Democrats foreign affairs spokesperson Layla Moran MP raised concerns about why the alleged hack had not been made public earlier.
"We need an urgent independent investigation to uncover the truth," Ms Moran said. "If it turns out this information was withheld from the public to protect Liz Truss's leadership bid, that would be unforgivable."
The government has refused to comment on any of the details reported by the Mail on Sunday.
"The government has robust systems in place to protect against cyber threats," a spokesman said. "That includes regular security briefings for ministers, and advice on protecting their personal data and mitigating cyber threats."
-- Courtesy of BBC News