Rose, who runs RBS’s commercial arm, is considered the preferred candidate, but chair Howard Davies has said the bank would consider external candidates.
It would make her the first woman to run any of the big four banks.
City and banking regulators are now in the process of considering RBS’s move to appoint her, Sky has reported.
An announcement of her appointment to lead the taxpayer-owned lender could come as early as this week.
The government still owns a 62.4% stake in the bank, as a result of its £46bn bailout of RBS during the financial crisis in 2008.
Rose, 49, has been at RBS for more than 25 years, mainly in a number of roles in its investment bank.
She was appointed deputy chief executive of NatWest Holdings, RBS’s non-ring-fenced bank, in December, and is also chief executive of commercial and private banking across the group.
She is a member of RBS’s executive committee, overseeing more than 12,000 staff, and is in charge of such brands as the private bank Coutts and asset finance provider Lombard.
She has previously been head of the bank’s Europe, Middle East and Africa operations.
She has in the past been shortlisted for the “most influential woman in investment banking” award by Financial News, and was included in the 2018 Women in FinTech Powerlist.
Rose is said to be “a passionate supporter of diversity” and is in charge of the bank’s employee networks.
She has also led a Treasury-commissioned inquiry – the Rose Report – into the barriers facing female entrepreneurs.
Rose appeared in Downing Street in July to launch the Investing in Women Code, a commitment to support the advancement of female entrepreneurship by improving female entrepreneurs’ access to resources and cash from the financial services sector.
In addition, she leads NatWest’s Entrepreneur Accelerator programme, an initiative supporting start-up businesses across the UK.
“Our next CEO will be confirmed in due course, once an appointment has been made,” RBS said.
Rose’s predecessor, New Zealander McEwan – who earned £3.6m last year – is leaving to join National Australia Bank,
Shares in the bank closed down 1.4% on Friday at 204.6 pence, giving it a market value of just over £25bn.