The Hammers have made Benitez their No 1 choice to replace David Moyes, while Russian giants Zenit St Petersberg have identified the 58-year-old as Roberto Mancini's successor.
But Balague claims Gazidis is in favour of appointing Benitez this summer.
West Ham announced the departure of Moyes on Wednesday and said a new manager would be hired "within the next 10 days".
It was previously reported that "Miners" suggested the coach to extend the contract for two years, but he is waiting for offers from clubs in the English Premier League.
Not including gate receipts, merchandise sales, shirt sponsorship and other deals that the club has, Newcastle will have received over £123m for finishing where they did in the league and the various other payments the league pays.
"Having taken stock of the situation and reflected now the campaign is complete", West Ham co-chairman David Sullivan said, "we feel that it is right to move in a different direction".
David Sullivan: 'West Ham United confident of appointing high-calibre manager'
It was in July of past year that Benitez urged Ashley to "keep his word" over transfer funds and reminded him that he could have quit the club for a mega-money deal in China.
"Now we've signed a new contract and I'm confident that new successes await us in the future". Hopefully Mike can preserve his phrase and we are able to do what we need to do.
We saw the obviously pre-planned statement that was released straight after the victory over Chelsea, once again offering "every penny".
'I might go away and make some cash.
'Then the assembly was essential to persuade me to remain. Mike told me "We have this money, it is available, it's OK", so, fine, we were happy with that'.
Sources close to the owner say he will give the boss at least £50m to spend this summer and perhaps as much as £70m, depending on player sales.