Kevin Lamour, who broke ranks with senior directors at world football's governing body last month and issued a scathing statement criticising Infantino over the plans, left his job as FIFA's chief operating officer on Monday.
UEFA has been a vocal critic of Infantino and England's Football Association has also withdrawn its support for the FIFA president.
"This is a problem of FIFA's making," Masters told the BBC.
"This is a self-inflicted wound. The organisation has pressed the self-destruct button, and the consequences of that flow now into the solutions that need to be found.
"I don't think FIFA should be ever contemplating selling off a stake in the World Cup.
"They're a not-for-profit organisation. They should really be holding that competition in trust for the future of football."
Masters added: "I think it's a bad idea, and the FA have taken the right position regards to Gianni."
As for a possible rival to Infantino, up for re-election next year, Masters said: "I think an emerging unity candidate should have on his or her agenda reform of FIFA, and perhaps even reform of the president's role and how it interacts with those things."