Scicluna will take up his new role at RSA on 1 January, and leave his non-executive post at Lloyds in March, following more than four years on the board.
It is thought that Scicluna will continue as chairman of property firm Great Portland Estates.
At RSA he will replace John Napier, who is stepping down after a decade at the helm.
Senior independent director Edward Lea said Scicluna will bring “considerable knowledge and understanding of the financial services sector and his board experience” to the 302-year-old insurer.
Chartered accountant Scicluna has spent most of his career at Deloitte, rising to audit partner and then chairman, a post he held for 12 years before retiring in 2008.
The 61-year-old Arsenal fan has a business degree and an honorary law doctorate from Leeds University, where a business school bursary is named after him. He is also a participant in Speakers for Schools, which links prominent public speakers with state schools.