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.
In a radio interview, Delaney said: "He is a serious candidate and he is one of the names being mentioned here (on Tuesday) morning.
"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.