Trapattoni's reign as manager came to an end by mutual consent earlier today after last night's World Cup qualifying defeat in Austria effectively ended their hopes of reaching next year's finals in Brazil.
Although the bookmakers have installed Martin O'Neill as a clear early favourite, Delaney has insisted that the FAI will take their time making their decision.
"Today was a day to deal with Giovanni and [assistant manager] Marco [Tardelli] in a dignified manner, and hopefully people will respect the job he has done," Delaney told Newstalk.
"We will meet and will determine a process. We do have time. The European qualifiers don't start until next September. We will take stock of the last couple of days and then discuss the process."
Click here to see our five early contenders for the job.