The businesses said they had reached an agreement that will see RBC paying 515p per share for the London-headquartered business. Brewin Dolphin's Scottish operation operated under the Bell Lawrie banner until 2009.
The takeover deal is more than 60 per cent ahead of where Brewin’s shares closed on Wednesday night, at 318p.
Founded in 1762, Brewin Dolphin has grown to become one of the largest wealth managers in the UK. It manages some £59 billion of assets on behalf of its clients.
If shareholders give the green light to the deal, which has been backed by Brewin’s directors, the firm will fall into the hands of the larger RBC.
It was founded in 1864 and has since grown to a market value of around £121bn, employing some 88,000 people worldwide.
Robin Beer, chief executive of Brewin Dolphin, said: "The Brewin Dolphin board is pleased to recommend the offer by RBC in the interests of our shareholders, our clients, our people and our business partners.
“Building on the strong organic growth that we have achieved to date, the combined businesses will create an attractive platform for future growth. As part of RBC we would be able to provide our clients with a broader range of products and services, and expand our distribution channels through leveraging RBC’s global presence.
“Our focus will be on maintaining continuity, so that we build on what we have already achieved.”
Doug Guzman of RBC added: “The UK is a key growth market for RBC, and Brewin Dolphin provides us with an exceptional platform to significantly transform our wealth management business in the region, giving RBC Wealth Management a number three market position in the UK and Ireland.
“By combining two highly complementary businesses, we will increase the depth and breadth of our services and position the combined business as a premier integrated wealth management provider to private and institutional clients.”