The playmaker has failed to live up to expectations at Anfield following his big-money move from AS Roma in 2009.
A series of injuries have seen the Italian make just 28 appearances for the club.
A move back to his homeland has been mooted, with Juventus and AC Milan, both of which he has spent loan spells with, believed to be interested in his services.
The arrival of Brendan Rodgers on Merseyside is thought to have given Aquilani hope of securing a first-team place with the club, and his agent Franco Zavaglia expects a decision to be made soon.
"We had a meeting with English executives but for the time being, nothing has emerged," Zavaglia told www.ilsussidiario.net.
"We expect a response from Liverpool which should arrive within the week."
Reports suggest the Reds may be willing to terminate his contract, making him a free agent - speculation Zavaglia denies.
"There is absolutely no possibility that Aquilani can become a free agent in order to then decide what to do," he added.
"I don't know if he wants to stay at Milan, we shall see. We just have to wait and see what happens and what Liverpool decides."
