Mr Morrison told reporters in Canberra on Saturday the meetings, set to focus on regional security, would be postponed.
He said his Indian and Japanese counterparts had passed on their sympathies to Australians affected by the bushfires.
"These events they're very aware of and completely understand," Mr Morrison said.
He said the talks with India and Japan on defence, security and intelligence arrangements were at an advanced stage.
"We were looking to progress those as well as broader trade discussions," he said.
In a statement, Indian Prime Minister Narendra Modi's office said he telephoned Mr Morrison to say he looked forward to hosting him in the future.
Mr Modi also passed on his condolences for the damage to life and property from the bushfires.
Japan's ambassador to Australia also passed on Japanese Prime Minister Shinzo Abe's sympathies for bushfire victims to Mr Morrison.
Mr Morrison said he would remain in Australia for the rest of January but hoped to reschedule the trips for early this year.
The prime minister faced criticism in December for holidaying in Hawaii with his family during the bushfire emergency.
