The appointment will be approved by the council on Thursday evening if none of the 15-members have raised any objections.
Typically, the Secretary-General has already informally consulted with council members before sending official notification, Reuters reported.
Griffiths, currently executive director of the European Institute of Peace, will replace Ould Cheikh Ahmed, who the United Nations said would step down after three years in the job when his current contract finishes this month.
A Saudi Arabia-led coalition intervened in Yemen in March 2015. Yemen, which relies heavily on imports for food, is on the brink of famine and nearly 1 million people have been infected with cholera.