Walsall chief executive Roy Whalley held talks with the Football Association of Ireland on Thursday to agree a compensation package.
"My discussions with (FAI chief executive) John Delaney were very amicable and we are very pleased with the way in which the FAI have dealt with this issue," Whalley told the Walsall website.
"I would like to take this opportunity to congratulate Steve Staunton when he is officially given the post and wish him and the FAI every success for the future."
Staunton has managed just 10 games for the Bescot Stadium outfit this season.