The Cherries have been linked with McKenna after manager Andoni Iraola announced this week that he would be leaving Dean Court when his deal expires at the end of June.
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Brazil said on talkSPORT: "Unless a big club come in, because we’ve got American backers and there is money there.
"They’re not bigger than Ipswich.
"Bournemouth? What have Bournemouth won? You’re having a laugh aren’t you?
"Look, honestly Ipswich are a bigger club than Bournemouth. I think Bournemouth have been brilliant, friendly club but no, no, no."
In their time, Ipswich have won an FA Cup, a top-flight league title and the UEFA Cup. Bournemouth, in comparison, remain trophyless at the elite level.