It is understood the brief is worth up to £300,000 in fees.
Drury beat three rivals to scoop the business, pitching against Irish agencies FCC Shandwick, Wilson Hartnell PR and Gallagher & Kelly PR.
All agencies involved in the beauty parade, which kicked-off earlier this year (PRWeek, 8 March), were previously retained on the bank's roster.
Bank of Ireland media relations manager Anne Mathews said the review reflects the bank's decision to consolidate external communications.
'We had a number of PR agency relationships within Ireland. One of the reasons we are undertaking this (review) is to get economies of scale and synergies of scale by having one single agency,' she said.
She hinted there could be further changes to the bank's UK roster, which includes Smithfield Financial, as part of the consolidation drive.
Drury reports to Mathews and head of group corporate comms David Holden.