Chelsea striker Fernando Torres is understood to be set for a return to Atletico Madrid, according to Sky sources.
Mark Hughes insists Stoke will not be looking to make any major additions in the January transfer window.
The new deal keeps him at Old Trafford until the summer of 2018.
The 18-year-old Belgian-born star has been in the headlines of late after scoring twice in United's 2-1 win at Sunderland a fortnight ago.
He had impressed during pre-season to win a place in United's side but his future had been called into question with his previous deal having been due to expire at the end of the season.
Januzaj, who joined the club from Anderlecht in 2011, told United's official website: "It's a great thrill to be able to sign for Manchester United. Since arriving here I have always felt like this was the right club for me.
"This season has been great for me - going on the pre season tour, making my debut in the Community Shield win at Wembley and then breaking into the team at Sunderland - it's like a dream. I want to work hard now and establish myself as a Manchester United player."
Manager David Moyes added: "I'm delighted that Adnan has committed his future to United. He is an outstanding talent and has impressed me consistently since I came to United in July.
"His balance, poise and ability to beat players is truly exceptional for a boy of his age. It's great that he recognises that this club, with its tradition for developing young players and giving them a chance on the biggest stage, is the best place for him to be."
Januzaj's extension gives Moyes' men a timely boost ahead of their Barclays Premier League clash with Southampton on Saturday.
A deal through to 2018 will also complete five years in England after the age of 18 for Januzaj, meaning he would have served the time needed to become available for the Three Lions on residency grounds, should he so wish.
When it was revealed the Football Association had asked after Januzaj in the wake of the Sunderland game, it kicked off a heated debate over eligibility.
Midfielder Jack Wilshere found himself at the heart of it, saying: "The only people who should play for England are English people.
"If you live in England for five years it doesn't make you English.''
Januzaj was born in Brussels and therefore eligible for Belgium but is also available to play for Albania, Kosovo, Serbia and Turkey.
Adnan Januzaj factfile:
1995: Born Brussels, Belgium, February 5.
2005: Signs for Anderlecht as a 10-year-old.
2011: Leaves Belgium to sign for Manchester United.
2013: Given his first squad number at Old Trafford, 44.
May 19 - Makes United's bench for the first time in Sir Alex Ferguson's last match in charge. An unused substitute in a 5-5 draw at West Brom.
July 29 - Rounds off an impressive pre-season tour under new manager David Moyes with a goal in a friendly win over Kitchee.
August 9 - Named as man of the match in Rio Ferdinand's testimonial against Sevilla.
August 11 - Features as a substitute in the Community Shield win over Wigan.
September 14 - Replaces Ashley Young against Crystal Palace for his Premier League debut.
October 5 - First Premier League start sees him score twice in a 2-1 win at Sunderland.
October 8 - Januzaj's brace sparks questions over whether he could one day be available for England selection under residency rules, leading Arsenal's Jack Wilshere to declare that "the only people who should play for England are English people''.
October 18 - Signs a new five-year contract at Manchester United.