If successful, the deal, which could be worth more than €120 million (A$180 million), would make Genoa the first club in Italy to be entirely owned by financial investors.
It would also mark the latest in a series of foreign takeovers of high-profile Italian clubs competing in Serie A.
Bologna-born Gallazzi, who controls London-based advisory firm SRI Group, has until September 30 to sweeten an initial bid which was submitted earlier this month.
Gallazzi, 53, has reportedly pledged to clean up Genoa's €110 million (A$165 million) debt and has asked a series of international hedge funds including UK-based Bybrook Capital to help fund his bid, which will cover all Genoa's debt and add about €10 million (A$15 million) in cash.
Bybrook, which received initial funding from U.S. giant Blackstone in 2014, declined to comment.
Gallazzi has been in talks with Genoa's owner Enrico Preziosi since the start of the summer and his bid on September 1 was rebuffed by Preziosi who publicly said it undervalued the business.
Preziosi, an Italian tycoon who controls one of the country's biggest toy manufacturers, Giochi Preziosi, resumed negotiations with Gallazzi on September 18, but the parties are still struggling to reach an agreement on price, with sources pointing to Preziosi's want for more cash.
Among the main sticking points is the valuation of Genoa's football players and its credits to other Serie A clubs, one of the sources said.
Gallazzi plans to finance his bid by raising equity and debt via GCFC, a UK-based financial vehicle created for this deal, which has Gustavo Perrotta, a manager of Mayfair-based merchant bank Hamilton Ventures, among its directors.
During Preziosi's 14-year reign with the club, Genoa have become a mainstay in the Serie A. But despite this achievement, the 69-year-old has continually reiterated his intention to sell.
Fellow Serie A clubs including AC Milan, AS Roma and FC Inter have also recently been taken over by foreign owners.
For Genoa, an eventual takeover by a UK-based financial entity would be something of a homecoming as the club was founded in 1893 by English doctor, James Richardson Spensley.