A 3-0 win over Borussia Dortmund on Wednesday put Juve into the quarter-finals, where they will face Monaco, as Allegri's side appear on course to convert their domestic dominance into European success.
With the Serie A leaders on course for a fourth straight Scudetto, Allegri also wants to end the club's 19-year wait for Champions League silverware.
"Juve's target should be establishing themselves as a permanent fixture of Europe's top eight," he said on Saturday.
"But first let's think about the Champions League, Serie A and the Coppa Italia.
"Perhaps there were other teams who breathed a sigh of relief at not being drawn against us on Friday.
"Monaco knocked out Arsenal and it will be a really tough quarter-final but we have a chance of going through.
"We played a great last 16 in the Champions League, attaining our target of going through."
With Roma playing later on Sunday, Juve could move 17 points clear at the top of Serie A when they host Genoa.
"Genoa will be a very difficult game," said Allegri before hinting at potential changes to his line-up.
"It's a great moment of form for us but we still have plenty of room for improvement.
"Tomorrow will be an important test to see how far we've come.
"At this stage of the season the players who have figured less are those who clinch the most important points.
"Roma have the chance to bounce back. 83 points are required to win the Scudetto."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.