The bid comes as Lloyds continues to struggle to conclude a deal to sell the branches to Co-op.
Co-op chief executive Peter Marks had cast doubt on the deal, citing “significant economic and regulatory issues”.
NBNK said it was proposing an “alternative demerger structure” where it would underwrite 100% of the package of branches being sold by Lloyds leaving Lloyds shareholders, including taxpayers, the option of receiving cash directly and/or receiving shares in a new, listed banking group.
A statement by Lloyds said it acknowledged receipt of a letter outlining an indicative revised proposal from NBNK.
However it also said: “We continue to negotiate solely with the Co-operative Group and are continuing to prepare for a divestment through an initial public offering.”
Sources close to the situation said the proposal by NBNK could value the Project Verde business at close to £2bn.
Lord Levene, chairman of NBNK, said: “NBNK offers none of the downsides to Lloyds of a standalone IPO. We have the right board, management, strategy and experience to run and grow a large scale banking operation and achieve our plans for Verde. Our shareholders are strongly supportive of our objectives.”
Gary Hoffman, chief executive of NBNK, said: “I believe we have tabled a compelling proposition that will invigorate competition, provide jobs and is the right solution for taxpayers.”