Allegri's men played out a 1-1 draw away from home in the first leg, but goals from Donny van de Beek and Matthijs de Ligt were enough to overturn Cristiano Ronaldo's opener in Tuesday's second leg, leaving the Italian champions out at the quarter-final stage.
The club were hoping for better this season, having landed Ronaldo from Real Madrid as they chased the elusive European crown, and the stunner at the hands of the Dutch side will not be taken lightly.
But Allegri's position with the club seems safe for the time being.
“I told the president that I will stay," he told Sky Sport Italia. "We'll sit down with the club and think about the future.”
Allegri also got the backing of club chairman Andrea Agnelli, who believes that the progress made by the club under the 51-year-old manager is too much to overlook on the basis of one failure.
“My job is to be there in certain moments and make general evaluations," Agnelli told Sky Sport Italia.
“Ajax fully deserved to qualify, but at the same time we are aware that for the last five or six years we’ve been a stable presence in the Champions League quarter-finals and so want to keep that progress going, year after year, year after year.
“Ajax are a club who work on cycles and this is one that started a couple of years ago. They deserved it, not just over the one match, but throughout the two legs. Of course, this is a competition where a good half-hour can make the difference between a trophy and elimination.
“We’re about to win the eighth consecutive Scudetto and that is extraordinary. Juventus were 43rd in the UEFA rankings and are now fifth, so we have to evaluate the whole situation.”
Allegri was also quick to praise Ajax's play, saying the Dutch club were deserving of their place in the last four, though he did take a bit of the blame for not making the correct adjustments for his side.
“We had done well in the first half, albeit a little too frenetic when we won back possession," he said. "Football is vicious, as you concede a fortuitous goal, the fear sets in, we got stretched out in the second half and should’ve had a different approach.
“Ajax fully deserved their result and qualification to the semi-finals.”