Australia sends asylum seekers to offshore detention centres and turns away boats carrying them.
A spokesperson for Mr Turnbull declined to comment.
According to a UN statement, Mr Ban and Mr Turnbull discussed several topics, including climate change and Syria.
Mr Ban addressed the topic of refugees using very direct language, expressing "concern" and asking that Mr Turnbull "reconsider" the policy.
"Noting Australia`s longstanding commitment to refugee resettlement, the Secretary General appealed to the prime minister to share responsibilities," the statement said.
""• Domestically, asylum is a hot political issue. Polls have shown that a significant number of Australians approve of taking a tougher stance.
• The two biggest rival political groupings adopted tough policies ahead of the September 2013 polls. The Liberal-National coalition, which won, had campaigned in part on a "stop the boats" platform.
• The government says the journey the asylum seekers make is dangerous and controlled by criminal gangs, and they have a duty to stop it.
• However, critics say opposition to asylum is often racially motivated and is damaging Australia`s reputation.
• Australia granted close to 13,800 refugee visas between 2013 and 2014. It granted about 20,000 visas between 2012 and 2013.
• Last Monday, a Syrian family of five became the first asylum seekers to arrive in Australia under a one-off plan to resettle 12,000 people fleeing the country`s conflict.
The comments followed reports the Australian Navy intercepted a suspected asylum-seeker boat near Christmas Island last Friday.
The immigration department did not comment on the incident.
The last documented asylum-seeker boat arrival to Australia was in 2014, when a group of 157 Tamil asylum seekers was intercepted north of Christmas Island.
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