The Electoral College, made up of 47 members, will meet today to vote on a confidential candidate list.
Candidates for election are nominated at the meeting and voted on by ballot. Any candidate receiving two-thirds of the votes of those present is declared Bishop-Elect. Otherwise, the College returns to the nomination stage and the cycle starts afresh.
Once a decision is made, the Cathedral is unlocked and an announcement made at the west door.
The election follows the retirement of Dr Barry Morgan who was head of the Church in Wales for nearly 14 years. Morgan split opinion by announcing his support for gay marriage as bishop.
The newly elected bishop will have 28 days to accept the offer and will be formally confirmed in April.
If he does bishop will be the 72nd Bishop of Llandaff.