Opposition parties in Ireland are calling for Gabriel Makhlouf's appointment to head the Irish Central Bank to be suspended.
But the Finance Minister in Ireland is not commenting on the investigation into the NZ Treasury Secretary over claims he misled the Government and unfairly smeared the National Party.
Makhlouf is scheduled to leave New Zealand on June 27 to take up his new position.
Ireland's Labour Party finance spokeswoman Joan Burton said the appointment should be put on hold pending the results of the investigation.
"The Governor of the Central Bank must be above reproach," Burton said in a statement.
"Following the collapse of our economy, public trust in financial institutions and the state bodies which govern them collapsed as well. It took the best part of a decade to get our economy and public confidence back on track.
"We cannot jeopardise that confidence by appointing someone under investigation in another country. [Finance] Minister [Paschal] Donohoe must now suspend the appointment of Mr Makhlouf until the full findings of the investigation in New Zealand are published."
Sinn Féin finance spokesman Pearse Doherty has made a similar call, while Fianna Fail finance spokesman has written to Donohoe to express concern, the Irish Times reported.
But a spokesperson for Donohoe would not be drawn on whether his view of Makhlouf's appointment had changed because of the investigation.
"This is a matter for the New Zealand Treasury," the spokesperson said.
"We have no comment to make on a political matter in New Zealand."
The investigation, headed by Deputy State Services Commissioner John Ombler, will look at Makhlouf's actions and public statements about the causes of the unauthorised access to Budget 2019 material, including the advice he provided to his minister at the time, his basis for making those statements and providing that advice, and the decision to refer the matter to the Police.
It will look to conclude before June 27.
National Party leader Simon Bridges has called for Makhlouf and Finance Minister Grant Robertson to resign for what he called unfairly smearing the National Party by implying that it had hacked Treasury's computer system.
National staffers had simply found confidential Budget information by using the Treasury's website search function.
The Treasury referred the matter to police, but police said that nothing unlawful appeared to have taken place.
The State Services Commission is already holding an inquiry into the security of the Treasury's website.