Trapattoni, 74, and assistant Marco Tardelli ended their five-year tenure in the job on Tuesday morning by mutual consent after the Republic's 1-0 World Cup qualifying defeat in Austria made it virtually impossible for them to reach next summer's finals in Brazil.
Former Reading manager McDermott, 52, whose parents are Irish, publicly stated last year that it was his lifetime's ambition to one day take on the Republic of Ireland job.
He was installed as Leeds boss in April, but only five months after replacing Neil Warnock at Elland Road has emerged as a serious candidate to replace Trapattoni.
"He is a serious candidate and he is one of the names being mentioned here (on Tuesday) morning," Delaney told BBC Radio Leeds.
"I am looking at the names being mentioned and I think it will come down to the three leading the market.
"I am looking at Mick McCarthy and Brian McDermott and I know that both men, having spoken to them over the past couple of years, they would both like to get their hands on this job, not necessarily at this stage.
"I know Brian McDermott, given his heritage with his parents being Irish, and having spoken to him on Irish radio, he would love to manage the Republic of Ireland someday."
McDermott steered Reading to promotion from the Championship in 2012 in his second full season in charge at the Madejski Stadium, but was sacked the following March.
Delaney also told Irish radio show Newstalk: "The board will meet within the next week. Today was a day to deal with Giovanni and Marco in a dignified manner and hopefully people will respect the job he has done.
"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.
"I think there will be plenty of interest, because first of all, 24 teams go to the European Championships and it's genuinely accepted we have a good crop of young players.
"It will be interesting to see who puts their best foot forward. I think names like Mick (McCarthy), Brian McDermott, Chris Hughton, Roy Keane all come into the pot, Martin O'Neill of course.
"Mick has done a very good job for Ireland and in England as well. We have to look at who can get the best out of the players available to us."