The Tigers are confident of sealing a deal to bring the Egyptian striker to the KC Stadium for the rest of the season within the next couple of days.
Zaki spent last season on loan at Wigan Athletic but, after an initially encouraging start, he fell out with Steve Bruce, leading the then-Latics manager to label the striker a "nightmare to deal with".
The 26-year-old returned to Zamalek but has since been linked with a host of clubs including Stoke City and Blackburn Rovers plus the Turkish trio of Galatasaray, Fenerbahce and Besiktas.
It seems, however, that Hull have stolen a march on their rivals with the player announcing on his personal website his intention to join the Yorkshire club, subject to passing a medical.
Tigers manager Brown, who met Zaki on Saturday night for talks, said: "Every time I have brought a player to the football club there are question marks, but I like sitting down with players to see if they have got a point to prove.
"I think Amr has probably looked at what he's done at Wigan last year and thought he would get 20 offers from the Premier League this year.
"He hasn't had those offers and he's had to go back to Egypt and dust himself down. Hopefully, we might be getting a player who is looking to prove a point."
Zaki is the overwhelming favourite to join the Tigers and was believed to be undergoing a medical yesterday.
However, talks are believed to have taken place with a number of possible targets in recent days, though any move for Vladimir Weiss seems to be on the back-burner due to Manchester City demanding a substantial loan fee.
On the Slovakian winger, Brown said: "I just want a little bit of sense with regards to loan fees, they're becoming part and parcel of the game.
"It is a ridiculous loan fee that they are after for a player that is unproven."
If Hull can sign Zaki this week, the Egyptian who netted twice in Wigan's 5-0 win at the KC last term will join a team with renewed confidence following Saturday's draw at Spurs.