Police stand near tributes outside a mosque which was attacked in New Zealand.
"Today GMP became aware of a post on social media making reference and support for the bad events in New Zealand", the Greater Manchaster police said in a statement.
The man "aged 24 from the Oldham area is now under arrest on suspicion of sending malicious communications", it added.
A GMP spokesman said: "This is a very hard time for people".
Greater Manchester Police said that officers made the arrest after becoming aware of a post on social media that "made reference and support for the awful events in New Zealand".
"Many people are in deep shock and are anxious", he said.
"Where the law permits and people cross the line, we will take robust action, which may include arrest and prosecution".
Fifty people were killed in the shootings at two mosques in New Zealand on Friday.
Brenton Tarrant, 28, from Australia, is alleged to have live-streamed the terror attack in Christchurch and outlined his anti-immigrant motives in a manifesto posted online.