The new prime minister has been in discussions with Mr Abbott after leaving him out of his new-look cabinet, in an effort to heal the wounds of last week's damaging leadership spill.
"It's not exactly clear what he's offered," Mr Abbott said outside his Sydney home on Monday.
It's believed the MP for Warringah is concerned about the impact of the role and how it sits alongside of, or interacts with, the Minister for the Indigenous Affairs Nigel Scullion.
Ex-Nationals leader Barnaby Joyce was on Sunday made special envoy for drought assistance and drought recovery.
Mr Morrison said his new ministry - which includes some of those behind the toppling of Malcolm Turnbull - would bring healing to the coalition and prepare the government for an election due by May.
New Liberal deputy Josh Frydenberg said finding a role for Mr Abbott would be of benefit to the Morrison government.
"(Indigenous affairs) is something that Tony's been passionate about for a long period of time. I do think he has something to bring to the table," he told 3AW radio.
Mathias Cormann, a supporter of Peter Dutton in last week's leadership spill, spoke positively about the possible role for Mr Abbott.
"The prime minister recognises that Tony Abbott has a strong contribution to make as a former prime minister in particular, and no doubt if any decisions are made in relations to these matters they will be announced in due course," the senior frontbencher told ABC radio on Monday.
Stuart Robert, who has been returned to the frontbench after a two-year absence, said Mr Abbott was "by far and away the best choice" to be the new indigenous envoy.
The former prime minister was passionate about indigenous affairs, and had shown his commitment to reconciliation when he was leader, he said.
"I can't think of a better person ... than Tony Abbott," he told Sky News.
The Liberal leadership turmoil has smashed the government's standing in the latest Newspoll, which showed Labor now holds a 56-44 per cent lead.
The government trailed by 51-49 just a fortnight ago.