Jones sought to ensure players also saw the bigger picture regarding the business side of the situation, including concerns over TV ratings and sponsors, the source said.
No Cowboys player has kneeled during the anthem, although the team met Wednesday to discuss Jones' comments.
While most Cowboys declined comment after the meeting, defensive tackle David Irving - the only player, along with defensive end Damontre Moore, to continue an anthem protest by raising a fist after Jones gave his ultimatum on Sunday - suggested his employer's opinion may win out.
Local 100 announced its complaint on Facebook, writing that the union aims "to stop owner, Jerry Jones, from threatening players with benching or termination for exercising their right to concerted activity under the National Labor Relations Act". Some requirements are that it must be played before every National Football League game, all players must be on the sidelines during the anthem and they should all face the flag with helmets in their left hand.More news: UCLA Basketball Recruiting: LaMelo Ball pulled out of Chino Hills High School
When asked if he would really sit a player like Dak Prescott, Ezekiel Elliott and Dez Bryant, Jones initially deferred. "We are trying to send Mr. Jones a message that there is a law here".
"Yesterday's meeting was about Jerry stating his position, and I think it was well received", the head coach said. I do believe we're united on this.
As the media entered the locker room on Wednesday, a song by YG and Nipsey Hussle came through Scandrick's wireless speakers.
Hill, who last month rankled ESPN execs and others over a tweet where she called President Trump "a white supremacist", was suspended Monday for writing on Twitter that people should put pressure on the Cowboys by "not patronizing" the team's advertisers.
"The word that comes to mind, and I don't care who doesn't like me using it, is "plantation", he said.