Conte's men saw off Getafe in the last 16 and Leverkusen completed the job against Rangers to set up an intriguing last-eight tie on neutral soil in Dusseldorf, which must be settled on the day.
Inter may have finished just a point behind champions Juventus in Serie A, but Conte has made it known that he is not entirely happy with the financial support he has received.
Tension appears to be building behind the scenes and there was talk ahead of the one-legged tie with Getafe that the former Chelsea boss may well walk this summer.
Those rumours have been put to one side for now, although it is clear that there is pressure on Inter - the favourites of many - to go all the way and lift the trophy on August 21.
Regardless of the circumstances, Inter's quality told in the end and they are now into their first European quarter-final since the 2010-11 Champions League.
They lost on aggregate to Schalke 04 that year and have not reached the semi-finals of a European tournament since 2010 - the year they went on to win UEFA's flagship competition.
Inter will carry a lot of momentum into this clash with Leverkusen, too, having now won four in a row in all competitions and gone nine games without a defeat.
Leverkusen's end-of-season form has been a little patchy, by comparison, with Peter Bosz's side failing to win back-to-back games since before the lockdown.
They finished the Bundesliga season in fifth, meaning that they will only qualify for next term's Champions League by going all the way in the Europa League.
Die Werkself also lost 4-2 to Bayern Munich in the DFB-Pokal final, though they soon put that defeat behind them in overcoming Rangers 1-0 on Thursday to seal a 4-1 aggregate win.
This will be Leverkusen's first European quarter-final since the 2007-08 UEFA Cup, while they have not reached the semis since 2002 when losing in the Champions League final.
Inter Milan Europa League form: WWW
Inter Milan form (all competitions): DDWWWW
Bayer Leverkusen Europa League form: WWWW
Bayer Leverkusen form (all competitions): WDWLWLW
Alexis Sanchez dropped to the bench in place of Lautaro Martinez in Inter's only change against Getafe, though the forward's loan move from Manchester United has since been made permanent and he may return here.
However, given his side have had five days off between games, Conte may be tempted to stick with the same XI that beat Getafe in the one-legged affair.
Eriksen is among those pushing for a recall, but he has not entirely won over Conte and Roberto Gagliardini appears more likely to start in a three-man midfield.
After his opener against Getafe, Lukaku has now scored in each of his last eight appearances in the Europa League (11 goals in total), a run stretching since 2014-15 with Everton.
As for Leverkusen, they had a highly-rated talent of their own in their ranks in the form of Kai Havertz, who has been linked with every big-name club across Europe this summer.
Havertz was unable to add to his tally of goals or assists against Rangers, though Moussa Diaby did by scoring the winner on the day and will surely retain his spot out wide here.
Inter Milan possible starting lineup:
Handanovic; Godin, De Vrij, Bastoni; D'Ambrosio, Barella, Brozovic, Gagliardini, Young; Martinez, Lukaku
Bayer Leverkusen possible starting lineup:
Hradecky; L.Bender, S.Bender, Tapsoba, Wendell; Aranguiz, Palacios; Diaby, Havertz, Bailey; Volland
We say: Inter Milan 2-0 Bayer Leverkusen
Inter have won both previous competitive meetings with Leverkusen, coming in the 2002-03 Champions League group stage, and are the strong favourites to win this latest encounter. Conte's men have hit a good run of form and, while they were not always at their best against Getafe, they know how to get the job done even on an off-day.