The 21-year-old, who played a key role in the 28-20 Premiership final win over Exeter on Saturday, said the players will be raring to go.
"If you need to get up for it, I think you are in the wrong sport because playing for England is an honour. Whoever is in the 23 that is picked will be raring to go," Itoje told British media.
"They (Saracens' team mates) never want to take a backward step in anything that they do and whether they have just won or lost, the hunger is always there."
Saracens became the first English club to do the European-domestic double since Wasps in 2004, and Itoje, who was named as European player of the year, was optimistic about the club's future.
"It means a lot to win the double. There's a bit more to come from this club," Itoje added.
"The core group of the squad is going to stay together for a few years to come and I think we can get better. This has almost been like a seven, eight-year project to get to this stage."
England play the first test against Australia on June 11 in Brisbane.
(Reporting by Nivedita Shankar in Bengaluru, editing by Ed Osmond)
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