Manchester City would be interested in signing Raheem Sterling if he were to become available, according to Sky sources.
Owls chairman Milan Mandaric made the announcement after the club had secured a place in the fifth round of the FA Cup with a 2-1 win at Rochdale.
Former Southampton and Northampton boss Gray, who has signed a two-and-a-half year contract, moved up from his assistant's role at Hillsborough when placed in temporary charge following the sacking of Dave Jones in early December.
The 53-year-old has steered Wednesday to six wins and four draws in his 12 games in charge, including their biggest-ever win against Yorkshire rivals Leeds earlier this month.
Mandaric said in a statement on the club's official website: "I'm very happy to say that Stuart has been appointed head coach of Sheffield Wednesday.
"When he was initially asked to step in his brief was to get us out of the bottom three, get some wins and clean sheets, and make some tangible progress on the pitch.
"Stuart has achieved all those things, he was given the chance to impress and has taken that chance with both hands."
Former Portsmouth and Leicester chairman Mandaric outlined his long-term plans for the club, who have been outside the top flight for nearly 14 years. They have also had two spells in the third tier since relegation from the Premier League in 2000.
Mandaric added: "I'm trying to implement a modern type of football management and structure of the club.
"In principle, Stuart's title allows him to do the same job, he has full power and responsibility, but eventually I would like to introduce a director of football to partner with Stuart and his staff.
"I'm looking at the bigger picture for where this club needs to go and in a modern football environment for the long-term benefit of Sheffield Wednesday.
"This is a big club that needs to go places and this will modernise the structure of the football side."
Mandaric gave Gray an extended period to try and make the job his own before Christmas after he made public his desire to consider former Sheffield United boss Neil Warnock for the job, only to change his mind following an outcry among Owls supporters.
Gray becomes the club's sixth permanent manager in eight years and fourth under Mandaric.