Canberra's hard-line immigration policy sees asylum seekers intercepted at sea trying to reach Australia sent for processing to camps in Papua New Guinea and on the South Pacific island of Nauru.
While aid agencies have called for the removal of the more than 1,000 refugees held for more than five years, Morrison is under immense pressure to resettle 40 children currently held on Nauru amid warnings they are suffering from declining mental health.
Detailing the first public timetable for the removal of the children, George Brandis, Australia's High Commissioner to Britain, said Canberra will remove the children, though he did specify where they would be moved to.
"There are hardly any children in Nauru and in New Guinea, and we expect that by the end of this year there'll be none," Brandis told British radio station LBC on Tuesday.
By resettling the child refugees, Morrison will meet a key demand from several independent lawmakers for supporting for his government.
Morrison's ruling Liberal Party is on course to lose its one-seat parliamentary majority as voters in a Sydney electorate look set to select an independent lawmaker to replace former prime minister Malcolm Turnbull.
Counting in the by-election continues, with independent Kerryn Phelps ahead of the Liberal Party candidate by nearly 1,800 votes, with just 2,000 ballots to be processed, the Australian Electoral Commission said
Turnbull quit politics shortly after he was removed from the office of prime minister in August following a party revolt that installed Morrison as leader.
The loss of Turnbull's affluent Sydney constituency would leave Morrison's government dependent on the support of five independent lawmakers to pass legislation in the lower house of parliament.
Several independent lawmakers have said they may back no confidence motions that would trigger an early election unless the children held on Nauru were removed. (Reporting by Colin Packham; Editing by Hugh Lawson)
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