The Metropolitan Police force said the men turned themselves in at a police station late Monday and were arrested on suspicion of a public order offence after allegedly creating a copy of the fire-ravaged west London public housing tower.
It is understood that all five men were being questioned at Croydon custody centre.
A video being shared on social media showed a cardboard and paper model of the lost tower being set on fire and crowds of people can be heard laughing.
Khadijah Mamudu, whose mother and younger brother escaped the fire, told the Press Association: "Their actions upset so many people around the world and they need to think long and hard".
He added: 'A total of 71 people died in the fire at Grenfell Tower, so many people lost so many loved ones, and many more have been deeply affected.
Undated grab taken from a video posted on social media of a model of Grenfell Tower being burned over a fire.
"I know that the police have been made aware of this video and will work to establish whether any offences have been committed".
In the clip posted online, raucous laughter can be heard off-camera as the model is set alight.
At the end of the clip, someone else can be heard saying: "That's what happens when they don't pay their rent".
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British Prime Minister Theresa May has criticised the "utterly unacceptable" video.
A total of 72 people died in the fire, which engulfed Grenfell Tower, a west London block, on 14 June past year. To mock that disaster in such a crude way is vile.
The video ends as the model is completely consumed by the fire, with one bystander saying: "Perfect".
Communities Secretary James Brokenshire said: "At a time when the bereaved and survivors are giving testimony to the Grenfell inquiry and reliving the unimaginable horror of that tragedy, it beggars belief that anyone should do this".
The Justice4Grenfell campaign group, which includes survivors and families of the dead, said it was "shocked at the inhumanity and callousness" of those involved.
"This was an unnecessary sickening act of hate against those who, through no fault of their own, have experienced the worst since 14 June 2017".
Natasha Elcock, from Grenfell United, said: "It's a disgusting video".
In a statement, London's Metropolitan Police Service called the video "vile" and urged people with information to come forward.