The 48-year-old, who has signed a three-year deal, assumes control of the Royals with the Championship winners in 19th place and seven points from safety.
Adkins guided the Saints to back-to-back promotions before being sacked in January and McDermott’s replacement is optimistic Reading can still beat the drop.
“We’ve got to believe we can do it. I want to be in the Premier League,” said Adkins.
“I’m excited about the future. I’ve had two weeks off. I am passionate about football, they are fighting, they have a good spirit and I’m looking forward to Arsenal on Saturday.”
He added: “We will scrap, it’s about the here and now. We have to concentrate on the Premier League.
“The players here have unbelievable spirit and they’re a good group of players. The Premier League is a great place to be.”
Adkins also revealed he spoke to former Royals boss McDermott about the club before agreeing to become Reading’s new manager.
“Brian has told me this is a great football club, with talented players and a great spirit. He is a top, top man, I had to think long and hard but I’m coming to a club with great spirit.”