Abbey National has agreed to sell First National for £848 million to General Electric.
The business will be bought by GE Consumer Finance, the consumer credit services business of General Electric.
About 1,400 staff are employed in the business being sold, based at First National's head office in Harrow, as well as other bases including Enfield, Caerphilly and Sheffield.
The deal does not include First National's litigation funding and motor finance businesses, which employ about 920 staff.
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Abbey National bought First National in 1995 for £285 million and put it up for sale in July as part of its plan to sell non-core assets. It has been seeing bidders for the business since September.