Mr Stevenson, who is currently head of European Investment Banking at Credit Suisse, was an adviser to The Treasury during the 2008 RBS bail out.
Any appointment would need to be rubber stamped by the Prudential Regulation Authority.
The experienced New Zealand born Mr Stevenson shares the same nationality as current RBS chief executive Ross McEwan.
The bank has been searching for a new finance director since Nathan Bostock announced in the autumn that he planned to resign.
l Another finance company expecting change at the top is pay day lender Wonga.
Several reports said Errol Damelin, co-founder and chairman of the business, is expected to step down in the coming weeks ahead of an expected Financial Conduct Authority clampdown on the short-term loans sector.
l Rail operator Arriva is trying to tempt some French local authorities into running private services, according to The Sunday Telegraph.
The company hopes to make some inroads into France by 2016 ahead of a planned shake-up of European railways to open them up to greater competition by 2019.
l The Sunday Herald reports that Glasgow School of Art's Centre for Advanced Textiles has invested £230,000 in state-of-the-art printing and steam fixing equipment.
The centre hopes this will help it to double its £300,000 annual sales as well as reducing cutting times and the need to hold large levels of stock.
Sticking with clothing the fashion tycoon Sir Philip Green is said to be aiming to break into the Indian market for the first time.
l The Mail on Sunday suggests that the Arcadia Group is making a global push on a number of its brands.
As part of that a deal has been sealed with Indian online retailer Jabong to stock a range of Dorothy Perkins products.
Alongside that Arcadia is said to have held talks with other other companies about launching Topshop and Miss Selfridge ranges in India.