Having previously performed the role throughout the north of Scotland, Mr Blair will now expand his remit to cover the whole of the country.
By leading a team of 100 employees, he will be tasked with providing financial support to Scottish firms with turnovers of no more than £25 million.
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Mr Blair reiterated his commitment to nurturing homegrown businesses of all shapes and sizes.
He said: "Having worked with Scottish SMEs for more than a decade I've seen first-hand how, with the right support, even the smallest businesses can grow to create employment opportunities and play a significant role in supporting the Scottish economy.
"Scotland has a wealth of entrepreneurial talent and Bank of Scotland remains committed to providing the financial support and guidance the country's SMEs need to capitalise on improved market conditions."
Alasdair Gardner, the bank's regional managing director for mid-markets and SME banking, said the appointment would benefit SMEs across Scotland.
He said: "With close to 30 years' experience working with Scotland's business community, Graham brings considerable knowledge and expertise to his new role.
"Scotland's SMEs are the cornerstone of Scotland's economy, a key driver of innovation and economic growth, and Bank of Scotland remains committed to supporting even more to realise their sustainable growth ambitions."