The acquisition is the fourth by Selbey Anderson since it was set up late last year, including the recent purchase of Greentarget.
Flagship Consulting is the second PR agency in the group, which now has 55 staff in five locations.
Flagship Consulting was set up more than 30 years ago by co-founders Jackie Murphy and Diana Soltmann. It provides integrated communications services to clients in the financial and professional services, social purpose, technology and travel sectors.
The agency has 15 staff and its clients include insurer AIG, Norweigian Crusie Line and Vote Leave.
Following the acquisition, Murphy and Soltmann will take on consultancy roles and deputy managing director Mark Pinnes, who joined the agency in 2014, steps up to become MD.
"Selbey Anderson’s thoughtful approach impressed us," Murphy said. "Their business plan is centred on helping creative agencies grow and so Diana and I felt that Flagship would be in good hands."
Selbey Anderson’s group chief executive Dom Hawes added: "Over the last thirty years, Jackie and Diana have built an agency that now has a tremendous reputation. We believe that Mark and his team have the drive and passion needed to further build on that legacy and take Flagship to the next level."