Serie A Preview: Fiorentina v Milan
The pressure on Inzaghi intensified after Milan were denied victory against Verona by a stoppage-time strike from Nicolas Lopez in last Saturday's 2-2 draw.
That was followed by reports during the week that the club had contacted Empoli with the aim of speaking to their coach Maurizio Sarri about taking up the role at the end of the season.
But Abate, who has completed his return from injury, remains right behind his manager and maintains Milan - who sit in mid-table - must target a European place.
"We are with our coach," he told the club's official website
"We are very close to our coach because he has always supported us.
"We have an obligation to go to Europe, starting from Monday. It will be difficult, but it would be wrong to think that we cannot get the three points."
Inzaghi's side have improved in recent weeks and are unbeaten in four league matches, but are still seven points behind their fifth-placed hosts going into this contest, having won just one of those four.
Fiorentina, meanwhile, are looking up and will have designs on reeling in Lazio and seizing third spot, which comes with the huge bonus of UEFA Champions League football.
Roma dented Vincenzo Montella's side's domestic prospects in their previous league match, ending an eight-game unbeaten run with a 4-0 success.
But Fiorentina responded with a solid display against Roma in the UEFA Europa League, finishing their home leg 1-1.
That result came at a cost, as midfielder David Pizarro suffered a torn hamstring and will therefore miss out, along with Mario Gomez, Joaquin and the suspended Nenad Tomovic.
There was some positive news for Milan on the injury front during the week as Stephan El Shaarawy saw a specialist, and his recovery from a broke metatarsal appears to be on track.
Andrea Poli is unavailable because of suspension, but Inzaghi has the option to make changes further forward with Keisuke Honda likely to come back in.