The Rossoneri take a two-goal lead to the Nou Camp, where bigger clubs than Milan have fallen foul of a Barca comeback, hoping to survive 90 minutes and join fellow Serie A side Juventus in the quarter-finals.
Allegri is fighting on two fronts, though, with 17th-placed Genoa threatening to both destabilise Milan's grip on third place and dash their hopes of an improbable late bid for the Scudetto.
The Milan boss told the club's official website: "Tomorrow [Friday] is not just a training exercise for the Barcelona game and I will pick the best team I can to see off the challenge of Genoa.
"They are in good physical condition and form and have picked up some good results recently.
"They will be trying to get something out of tomorrow [Friday] so they can move closer to their target of staying in Serie A. They have a lot of players with great technique and I think we will have to be patient and know when to go at them.
"We have to be wary of them while trying to get the three points that will help us defend third place until the end of the season.
"With all the talented guys I have here it's just not possible that we can allow ourselves an off-day."
Holland midfielder Nigel De Jong and defender Cristian Zapata are Allegri's only confirmed absentees, with striker Mario Balotelli likely to start on the bench as he recovers from a shin injury.
Genoa, meanwhile, are five points clear of the bottom three but suffered the first reverse of the second reign of Davide Ballardini courtesy of Roma last time out.
Following the 3-1 loss at the Stadio Olimpico, in which Francesco Totti scored his 225th Serie A goal, Ballardini told La Repubblica: "We are convinced that if we keep playing this way, we will end up satisfied.
"We can't relax yet because of the league table, but we can be confident of our own ability."
Ballardini is confined to the San Siro stands for remonstrating with the referee in Rome while midfielder Juraj Kucka serves a one-match ban, with Matuzalem, Antonio Floro Flores, Ruben Olivera and Thomas Manfredini all nursing injuries.