Tom Hicks is expected in London early next week to discuss the future of Liverpool with Dubai International Capital.
The club's co-owner has been in a stand-off with DIC all week over their attempts to buy Liverpool for £400million.
By the time Liverpool return from the San Siro, the future ownership of the club could be known - and a DIC source said: "We are in the end zone."
DIC is the financial arm of Dubai's ruler Sheikh Mohammed bin Rashid Al Maktoum, and Staveley reports directly to him and chief executive Sameer Al Ansari - a lifelong Liverpool fan who has been the driving force behind two attempts to buy the club.
DIC have been encouraged by claims that Hicks has agreed not to block the sale of his co-owner George Gillett's 50% holding.
Hicks has also intimated he is still not prepared to sell any of his stake - but DIC appear prepared to go with that suggestion, for now at least.
The source said: "To accommodate a deal, DIC, within reason, would accept certain agreements.
"It looks as if we are in the home straight. A deal involving both parties could be imminent."
After a week of constant cross-Atlantic negotiations through Staveley, lawyers and accountants, DIC are confident they will eventually control Liverpool.
It is also believed that Gillett, who has wanted 'out' for some while, is not willing to sell a small percentage of his stake to Hicks.