Reid, who won Young Irish Player of the Year last season, was the subject of a £1.5 million offer last month but this price was publicly ridiculed by Lions Chairman Theo Paphitis, who likened Graeme Souness' transfer dealings to someone who shopped at Freemans.
However, it is likely that Rovers' new bid will be more in line with Millwall's valuation of the player as the London side look to strengthen their side in time for the new season.