The 22-year-old is available on a free transfer after Telford went in to liquidation and Quinn is looking to bring the forward to Gay Meadow.
Grant, who came through the ranks at Crewe Alexandra, worked with Quinn when he spent a loan spell at Northwich Victoria in the spring of 2002.
Northwich had been keen to re-sign Grant but appear to have conceded defeat to Shrewsbury, who have reclaimed their place in the Football League.
Though Quinn has refuted suggestions a deal has already been arranged, he has spoken to Grant's agent regarding a possible move.
Quinn has already raided Telford once during the off-season after snapping up defender Stuart Whitehead not long after the club secured promotion back from the Conference.
