The 52-year-old was appointed by the Championship club in April and won three of his five games in charge to lift them clear of relegation danger.
McDermott has said that he is looking forward to the challenge that lies ahead at Elland Road.
"Expectations are what they are. The fans want expectations because this is Leeds United," he told the club's official website.
"I like the challenge and I like the people at Leeds and I really feel their commitment.
"I haven't been here very long but I really feel the commitment of the people. They really want the club to grow and get better because potentially we have an amazing club."
McDermott guided Reading to the Championship title in 2012.