Football Association of Ireland chief executive John Delaney had confirmed that McDermott was high on their list of possible replacements for Trapattoni, while the former Reading boss, whose parents are both Irish, stated publicly last year that it was a lifetime's ambition to take on the role.
McDermott spoke of his "immense pride" at having been linked with the post at a press conference on Thursday, but confirmed he would be staying put at Leeds, where he succeeded Neil Warnock as manager in April.
"My reaction is immense pride more than anything," McDermott said. "I was sat with my very good friend last night in my flat and we were looking at the Sky Bet odds - Martin O'Neill was top and I was second.
"If you consider I've been a manager now at this level for four years and to get to that, for me, there's nothing but pride.
"So I'm probably doing something right, which is good, and everybody knows what I feel about Ireland, my family, my history and how important that is to me.
"So those would be the words that I would use - incredible pride."
When asked if wanted the job, or would be offered the job, McDermott said: "No. My situation is that one day I want to manage the Republic of Ireland, but at this moment I'm at a fantastic football club.
"I've kind of been adopted here and feel I owe this club and I owe these supporters.
"They're really important to me. I came here in April and certain things were said, but I'm here now and I'm committed 100 per cent.
"I would regard Leeds United as a very big club. Very big clubs can spend £50million on players and we can't do that here, but we can find a way to make us successful.
"It's really important for me that we become successful at Leeds United and one day if I ever get the opportunity - and it's been a privilege to see my name bandied around - that (Republic of Ireland) would be a job that I'd really want to do.
"But at this moment I wouldn't be leaving Leeds United for any other club and in years to come when Leeds get fed up of me, that might be the case and if I got that opportunity I'd have to take it."
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, with Leeds owners quickly turning to the former Arsenal winger to try and get the club back in the Premier League after a 10-year absence.
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- Leeds United