Villa, who started the season brightly and were second in the middle of September, have slipped to 15th in the table, the same lowly position they ended last season.
Lambert's misfiring side have taken only 12 points from their last 18 league games, but Fox told BBC Sport sacking Lambert mid-season would be like 'flipping a coin'.
He explained: "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."
Since beating Leicester City on Dec. 7, Villa have gone seven league games without a win and scored only one league goal.
They have fared better in the FA Cup beating Championship sides Blackpool and Bournemouth, but Fox believes Villa's players are close to improving their form and that the statistics don't tell the whole truth.
"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," he added.
"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."
Villa, three points above the relegation zone, face a tough trip to Arsenal on Sunday. Arsenal, who are fifth, won 3-0 at Villa Park in September and have lost only once in their last 10 league and cup games.
(This version of the story was refiled to clarify headline
(Reporting by Mike Collett; editing by Sudipto Ganguly)
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