Amadi will join the charity in a new role called chief supporter officer, although his start date has not yet been decided.
He succeeds Mark Astarita, who left his position as the charity’s executive director of fundraising to join the consultancy Aldrich and Ward in November.
In a statement, the charity said the change of title for its top fundraising job "reflects the British Red Cross’s desire to offer a great experience that inspires supporters to connect with the organisation at different stages of their life in different ways".
He also chaired the Institute of Fundraising between 2008 and 2010 and founded the IoF’s Black Fundraisers’ Network.
The Red Cross is one of the UK’s largest fundraising charities, reporting a fundraised income of £137.8m in 2016.
Mike Adamson, the charity’s chief executive, said: "Paul is a hugely experienced and progressive fundraiser who embraces this new world of supporter engagement. Over his two-decade fundraising career, Paul has improved the performance of some of the UK’s most high-profile charities, taking responsibility for all aspects of fundraising.
"We estimate that up to five million people come into contact with the Red Cross each year. Each of them is a chance for us to connect and inspire people in ways that work for them at whatever stage of their lives they are at. That is the challenge and opportunity for our new chief supporter officer."
Amadi said he was "absolutely thrilled" to be joining the charity.
"For me it represents an opportunity to join an organisation of global renown and relevance in a role of critical importance and at a moment of change," he said.
"I look forward to building on the considerable success already achieved by a talented team and committed volunteers and increasing the British Red Cross’s ability to help people in crisis."
A British Red Cross spokeswoman was unable to confirm how much Amadi would be paid in the role. Astarita was paid £121,800 in 2016, according to the charity’s accounts.