New British banking venture NBNK said talks with Lloyds Banking Group over the possible purchase of 632 branches were going well, RTE Ireland reported.
It added that the UK's financial regulator was happy with its plans so far.
The sale could make up to £1.5 billion and create Britain's seventh-biggest bank, with a 4.6% share of current accounts and 5% of mortgages.
The UK market is dominated by four major players – Barclays , HSBC, RBS and Lloyds.
Lloyds, which is 40% owned by the UK government, ended a period of exclusive talks with The Co-Operative Group in April, paving the way for NBNK, which is run by former Northern Rock executive Gary Hoffman, as an alternative bidder.
The sale of the branches, code named Project Verde, was forced upon Lloyds by the European Commission as payback for the bank receiving state aid during the 2008 financial crisis.
"NBNK has been in discussions with Lloyds following its renewed offer for the Verde asset package. These discussions and meetings have progressed positively," NBNK said on 1 June.
NBNK confirmed that it had presented its plans to the Financial Services Authority, the UK's financial regulator, at the request of Lloyds.
Lloyds management had indicated it would not fully engage in discussions with NBNK unless it was confident NBNK could get regulatory approval.
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