National sevens squad member Rory Hughes, 20, is also the subject of a report to the procurator fiscal in relation to the incident, which is said to have happened last month at a kebab shop in west end of Glasgow.
Glasgow Warriors pair Grant and Maitland were accused just three days before they are due to line up for the Scots in the autumn Test encounter with South Africa at Murrayfield.
Alleged victim Ally Maclay, 29, who plays for Glasgow Hawks, was treated at the Western Infirmary.
And a police spokeswoman confirmed: “Three men aged 25, 28 and 20 are the subject of a report to the procurator fiscal as the result of an alleged serious assault and an assault.”
Another Scottish internationalist, Ryan Wilson, had been reported on identical charges eight days ago.
A Glasgow Warriors spokesman said: “We regard any allegations of player misconduct extremely seriously.
“As part of our own procedures we would always instigate an extensive internal inquiry into any allegations of player conduct falling below the high standards that are part and parcel of Scottish Rugby’s ethos and values.
“We are continuing our own internal investigation into an alleged incident last month, while at all times co-operating fully with the authorities.
“The club will be making no further comment at this time.”
A spokesman for Scottish Rugby said: “We have been kept fully appraised of the alleged incident and we will continue to monitor closely any developments.
“We cannot offer any comment on the alleged incident at this time. However, we do reiterate that it is imperative that the values of our game are upheld by all those involved with the sport.”