The bank, which is 84% state-owned, could opt to take control of Kop Holdings, the company used by Hick and Gillett to purchase Liverpool in February 2007.
Purslow, along with chairman Martin Broughton and commercial director Ian Ayre, form a majority on the board and could approve a sale without the American owners' agreement.
Hicks and Gillett, who took control of the club in February 2007, are believed to want £600m for the club, while a figure of £420m would effectively see them leave without having made a profit.
However, if the club was to fall into bank ownership it could be available for about £280m, a figure that includes £40m of penalty fees.
Meanwhile, Hodgson is left trying to turn round a club that finished seventh in the Premier League last season.
They have won only one of six league matches this campaign and were knocked out of the Carling Cup at home by League Two Northampton Town last Wednesday.
"The fans here are very passionate and care an awful lot about this football club," he added.
"You cannot criticise them for showing their displeasure about the situation."
Captain Steven Gerrard and Jamie Carragher applauded the fans who had remained in the ground as they returned to the pitch to warm down after the match.
Bookmark with:
What are these?