Mrs May and Cardinal Nichols worked closely together on the Modern Slavery Bill and she travelled with him to attend a meeting on the issue at the Vatican in 2014.
"I am personally delighted at your appointment," the clergyman wrote. "I know from the work we have done together that you have so many qualities to bring to the service of our countries at this time".
"I appreciate the maturity of judgement, the steely resolve, the sense of justice and the personal integrity and warmth you have always shown.
"I am only too aware of the magnitude of the undertaking you are accepting. I assure you of my prayers and those of the Catholic Community in England and Wales.
"As you take up this new and demanding office of Prime Minister, I assure you of my personal support and I look forward to working with you across a wide range of issues in service of the common good."
Theresa May is a practicing Anglican whose father was a vicar.