Santander chairman William Vereker said the newly appointed executive “brings a powerful combination of experience, knowledge and energy, twinned with a positive vision for the future of the bank”.
Last year Mr Regnier was reportedly one of the highest-paid bosses of a UK building society. He has been in the role since 2017, and will be replaced by Yorkshire's chief operating officer Stephen White, subject to regulatory approval.
Yorkshire Building Society chairman John Heaps said: "He has led the society superbly through a period of significant challenge as we have addressed both the transformation to a more digitised world and the complexities of the Covid pandemic."
Mr Regnier said: "I am honoured to be joining Santander, a bank with a rich heritage in the UK, a strong track record of innovation and a commitment to helping our country prosper.
"I am excited to be able to make a real difference to people and businesses across the UK, helping them to thrive and grow."
He added: "I have thoroughly enjoyed my time at Yorkshire Building Society and it has been a great privilege to have been CEO. I feel that now is the right time for me to take on a different challenge and leave the society in excellent shape for my successor to take the society on the next phase of its journey."