The Royals had three separate bids rejected for their former player and were willing to make the Iceland international their record signing, but were unable to strike a deal
The Iceland international, who came through the ranks at Reading, has struggled to hold down a first team place at Spurs following his move from Hoffenheim in the summer and McDermott has confirmed the club came close to resigning the player on Thursday night.
"We tried to bring him back," he told reporters. "We had spoken to Spurs and we made three offers.
"I thought at one stage we had a real chance of doing a deal, but unfortunately, we couldn't manage to do it.
"Gylfi is a Tottenham player, but we've know him since he was 13 years old and it would have been great to bring him back.
McDermott has worked hard this week in an effort to bring in some new faces at the Madejski along with director of football Nick Hammond and the club's owner Anton Zingarevich.
"I haven't had a lot of sleep in the last three days," he added.
"After the Chelsea I watched the tape back and went to bed at about 3.45am - I got up at 7am and then came to work. Then last night I left here at 11.15pm."