Mr Morrison also says the first of those found to be refugees could be resettled in Papua New Guinea in June.
The Minister yesterday held the first of what are planned to be monthly meetings about the offshore resettlement program with his Papua New Guinean counterpart, Rimbink Pato, in Port Moresby.
About 1,300 people are currently being held in the Manus Island centre.
The first group of asylum seekers was transferred there from the Christmas Island facility in August last year.
Mr Morrison says the Government is making "tremendous progress" in developing the offshore resettlement program with PNG.
"This was an arrangement that was frankly a blank sheet of paper six months ago and I think we're putting some real meat on the bone here," he told Sky TV.
In a joint statement, the two countries say the PNG cabinet has decided to allow "the first tranche of initial assessments to be delivered to transferees within a month" and that it was "anticipated the first refugees could be resettled in June".
The PNG government has also approved the creation of a refugee visa that will provide work rights and freedom of movement to asylum seekers found to be refugees.
The ministers confirmed that all refugees will be resettled in PNG - despite a statement from PNG prime minister Peter O'Neill last month that his country would take only "some" of those whose refugee claims are approved.
Mr O'Neill made the comment during an official visit by Prime Minister Tony Abbott, which was dominated by discussion of the offshore resettlement program.
"I don't know why that was ever in doubt. That's how this arrangement was to work," Mr Morrison told Sky.
But then he added that "to have that codified in that way is a major step forward".
The statement from Mr Morrison and Mr Pato also said two thirds of the asylum seekers on Manus Island have had initial interviews and that a review process for those asylum seekers not found to be refugees will be established by the end of this month.
The ministers also discussed the "urgency and need to complete" the investigations into the riot at Manus Island in mid-February, that resulted in the death of one asylum seeker and left dozens injured.
But Greens immigration spokeswoman Sarah Hanson-Young says the process with PNG is a "facade designed to reject as many refugees as possible".
The previous Labor government reopened the centre in November 2012 after closing it down shortly after coming to power in 2007.
It was first set up by the Howard government in 2001.