The Old Lady went into their quarter-final second leg in Turin on Tuesday tied 1-1 with the Dutch club following last week's first leg in Amsterdam.
Cristiano Ronaldo opened the scoring just before the half hour, but the Italian champions then lost focus and were eventually knocked out the competition after losing 2-1 (U.S. only):
Allegri's near five-season tenure at Juve has been successful as the Old Lady have reestablished their status as one of Europe's elite clubs, and they are on the verge of an eighth consecutive Serie A title.
But his future was questioned following Tuesday's Champions League exit:
However, the 51-year-old said, per Luke Augustus of MailOnline: "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."
And club president Andrea Agnelli backed up Allegri's assertion, per Sky Sport Italia (h/t Football Italia):
"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."
Ronaldo, 34, scored both of Juve's goals in the tie against Ajax, as well as the hat-trick that turned the last-16 tie against Atletico Madrid in Juve's favour.
As a five-time winner of the competition with Real Madrid and Manchester United, Ronaldo was brought to Juve to help them finally win the Champions League again following two final defeats in four years.
When asked after Tuesday's exit if the Old Lady had thought Ronaldo would fix all the club's European problems, Allegri said, per Augustus:
"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.
"I've always said that to win the Champions League, you need to reach a certain stage in good form and with good fitness levels. We've been in a bit of an emergency situation for a while and that can only be sustained for so long.
"We were missing four or five players tonight. Daniele Rugani and Leonardo Bonucci had good games for 50 minutes or so, but we lost our unity and organisation, giving Ajax the advantage."
Ajax were fully deserving of their victory after a brilliant performance, which followed a fine showing last week in the first leg.
Donny van de Beek, 21, and 19-year-old captain Matthijs de Ligt were Ajax's goalscorers on Tuesday, and both are products of the the Dutch club's famous academy.
Erik ten Hag's exciting young side have deservedly established themselves as the neutrals' favourites this season as they have stirred memories of great Ajax teams of the past:
They have now knocked Juventus and Real Madrid out of this season's Champions League, while also going unbeaten in two group-stage matches against Bayern Munich.