The Labour-led council has granted planning permission to the club to build a new stadium.
Mr Hammam, who made his name as chairman of Wimbledon FC's "crazy gang" during the 1980s, said Mr Goodway - and other councillors backing the stadium scheme - should be supported, although it makes no explicit mention of the June 10 poll.
The letter reads: "Councillor Goodway and his fellow members and officers have driven a hard bargain, these social and enabling works will cost a great deal of money, but above all they saw the opportunity and they have delivered.
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In a letter sent out to the thousands of Cardiff City Football Club supporters headed "about your council", Mr Hammam praises the council leader for his support for a new £40m stadium.
He states, "People have been very kind and say that I have achieved this but everyone should be under no illusion, the only people that have opened the way forward to Cardiff City Football Club is Cardiff County Council and its leader Russell Goodway.
"He has his detractors, those that would stop all development because of their negative petty politics, but he has not folded under the pressure and the way is now clear for us to deliver, this is fair."
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