He also threatened attacks in the UK, saying "brothers that I know there... are itching to do an attack."
"This is a direct threat," he added.
Communicating with the paper via the encrypted text message service Kik, Elmir also praised the gunman who killed 38 tourists, including 30 Britons, in Tunisia two weeks ago.
"May Allah bless the man who slaughtered those filthy kuffar [infidels] and May Allah grant him the highest level in Jannah [Paradise]," he said.
The Mail on Sunday said police were investigating the threat made by Elmir.
Elmir was working as a butcher in Sydney when he disappeared.
He surfaced later in Syria, where he appeared in an Isil video.
"Until we put the black flag [of IS] on top of Buckingham Palace, until we put the black flag on top of the White House, we will not stop, and will keep on fighting," he said in the video.
It appears he and Abase married in March.
News of the marriage comes as it emerged that Seiffedine Rezgui, the gunman who carried out the massacre in Tunisia, trained in Libya with Mohammed Emwazi, the British Isil fighter who has appeared in several of their executuon videos beheading Western hostages.
The Sunday People reported that Emwazi, also known a "Jihadi John", and Rezgui were at the same training camp in Sabratha near the Tunisian border at the same time.
