While Mehta has been arrested for drunken driving, Malik, who is not a part of the official contingent is facing a more serious charge of sexual assault.
A Scotland Police spokeswoman confirmed the arrests but refused to divulge anything more.
"On the August 2, a 49-year-old man was arrested following reports of an alleged assault at City Centre of Glasgow. On August 2, a 45-year-old man was arrested following reports of an alleged assault in the west of Glasgow," said the spokeswoman.
The duo was, however, not staying with the official 215-strong contingent which is putting up at the Games village. Both of them were reportedly lodged at a local hotel.
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Asked if Malik had been arrested for alleged sexual assault, the spokeswoman said, "I cannot confirm right now."
"We are aware of the matter. It is a matter of Scotland Police and any question related to this issue should be addressed to them," said Commonwealth Games Federation CEO Mike Hooper.
The Indian Consulate in Edinburgh said it is liaising with the police on the matter.
"We are also in touch with the team," he added.
According to unconfirmed reports, Mehta was allegedly caught drink-driving and told the police to not release details of his arrest even to his family, which is travelling with him.
Depending on how over the legal alcohol limit he was, Mehta could face anything from a hefty fine or a driving ban to a jail term.