Dallas Cowboys: Huge Sponsor Stands By Team ... Despite Greg Hardy Pics

Seems the Greg Hardy photos aren't scaring off one of the Dallas Cowboys' biggest sponsors -- with American Airlines vowing to stand by the team ... despite the controversy.

Critics of the Cowboys are calling for sponsors to pull out -- hoping to put pressure on Dallas to fire Hardy after photos surfaced which appear to show the brutal injuries he inflicted on his ex-girlfriend.

But American Airlines -- which has a partnership with the Cowboys until 2019 -- tells TMZ Sports they're not running for the hills.

"American Airlines supports teams, not individual players," an AA spokesperson tells us.

"We have a huge employee and customer base in Dallas/Fort Worth and we partner with the Cowboys on several community projects throughout the year. As part of our initiatives to improve the communities we serve, we partner with several organizations that support and protect victims of domestic violence."

However, our sources tell us ... when Hardy was signed to the team, AA discussed the situation internally and expressed concerns about being connected to the embattled player.