Juve's quest for European glory was put on hold for another season following Tuesday's shock 2-1 second-leg loss at home to Ajax.
Cristiano Ronaldo's opener was cancelled out by Donny van de Beek and Matthijs de Ligt, who led Ajax to a memorable 3-2 aggregate victory.
However, Allegri is set to remain in Turin after telling Sky Sport Italia: "I met the president a couple of days ago and told him that I will stay at Juventus.
"We'll sit down with the club and think about the future."
Juventus president Andrea Agnelli also put his faith in Allegri moving forward, adding: "We've said since the summer that the Champions League is an objective and will be every year. This was a relatively young starting XI tonight, so it can grow with Allegri on the bench.
"We'll sit down at the end of the season to discuss the contract, as we said six weeks ago – and we have a club that guarantees the future and continuity of a prosperous future for Juventus."
Five-time Ballon d'Or winner Ronaldo, who also scored in the first leg, gave Juve a 28th-minute lead with his 126th Champions League goal.
However, Ronaldo's semi-final run – dating back to 2009-10 – came to an end as giant-slaying Ajax produced another upset, just like they did against reigning champions Real Madrid in the last 16.
Asked if some of his players felt Ronaldo could resolve everything, Allegri – who was without the likes of Giorgio Chiellini, Mario Mandzukic and Douglas Costa – replied: "No, absolutely not. Ronaldo has given us a lot over the course of the campaign, but when you reach the quarter-final, you need every player.