Last week, Lloyds ended its exclusivity agreement with the Co-operative Group over the sale of the so-called Verde portfolio, which includes the 185-strong Lloyds TSB Scotland estate.
NBNK, whose chief executive is former Northern Rock chief Gary Hoffman, held discussions with Qatar Holdings and Abu Dhabi's Mubadala fund.
Meanwhile, Lloyds is pressuring NBNK, whose directors include Lord Forsyth and Lord McFall the Labour peer who previously chaired the Treasury Committee, to demonstrate it would be permitted to buy the network.
It was reported that NBNK has been told to produce written evidence from the Financial Services Authority that it would get regulatory clearance for the deal.
Lloyds is keen to ensure NBNK has sufficient capital strength, appropriate governance structures and a suitable technology platform.
NBNK is also understood to be interested in Glasgow-based Clydesdale Bank, which is undergoing a strategic review.
Lloyds, which is 41% owned by the taxpayer, has to sell the Verde portfolio by November 2013 as a condition of European Commission approval of its Government bail-out.
The Co-operative remains Lloyds' preferred bidder.