Sources added that Morata, who is on loan at Juve from Atletico Madrid until the end of the season, was open to a return to Spain, but the Italian club don't want to lose him.
"Morata stays here," Allegri confirmed in a news conference on Wednesday. "He is not leaving this month. We had a face-to-face meeting and I told him: 'You're 100% staying.'"
Juve have an option to make Morata's move permanent in the summer, but sources told ESPN the Serie A club were open to letting him end his loan in Turin early.
However, the lack of possible replacements available has forced them to change their opinion after initially offering Barca optimism that a deal might be possible.
Former Real Madrid striker Morata, who has scored seven goals in 23 appearances across all competitions for Juve this season, is contracted to Atletico until 2023.
Barca coach Xavi Hernandez has expressed a desire for the Catalan club to sign a No. 9 after finalising a deal for versatile forward Torres.
The club also sounded out a deal for Edinson Cavani, who sources have told ESPN would have been interested in moving to Camp Nou on an 18-month deal, but new Manchester United coach Ralf Rangnick wants the Uruguay international to stay at Old Trafford.
Any new signings Barca make, though, are dependent on departures first. They are not yet even able to register Torres with LaLiga until they make more room on the wage bill.
"We made an exception with Ferran," director of football Mateu Alemany explained this week. "But any other [January] signings will be dependent on moving on other players first."