Villa have slipped to 15th in the table following a run of seven top-flight games without a win - scoring only one goal in the process.
That dismal run has led to fans calling for Lambert to be sacked, as well as venting their fury at owner Randy Lerner, but Fox is in no doubt that the Scot is the right man for the job.
He told BBC Sport: "When things aren't going well, fans bay for that type of blood,
"That's not the way that I or the owner are going to make a decision. It's a false narrative."
He added: "I think football's a funny business and I think to put it all on the manager, again, that's a bit of a false narrative.
"There's a lot of things happening at this football club, and clearly what happens on the pitch is a focal point.
"Clearly the manager is the one person who can really influence that, so everyone focuses on it.
"Our focus for the last couple of months has been on making sure we support Paul to make us as successful as possible, and I think we're heading in the right direction."
Lerner revealed his intention to sell the club back in May, but remains at the helm eight months on and Fox said his fellow American is still committed to Villa.
Fox said: "He's very involved in the football club, he watches every match,
"I've been at his house watching matches together and both of us are standing up pacing around at various points of time - depending on what's happening - so he knows what's happening here.
"He's very, very well informed as to the business side of what's happening on the pitch,
"I don't think he's a motivated seller right now. I think he'll want to make sure that he's put Aston Villa on the best possible path for the future."
