Goodell got a question Sunday about the latest debate over the name during a question-and-answer session with Dallas season ticket-holders before the Cowboys' game against the Redskins.
The commissioner repeated what he said during the owners' meetings last week — that he grew up in Washington and considered everything about the name to be something that honoured the tradition of Native Americans.
"But I think we also have to be sensitive enough to at least listen and try to see what it is we can do if we're insulting any element of our fan base, or non-fan base for that matter," Goodell said.
"I think Dan Snyder is way down the road on doing that. I'm confident he's listening. I'm confident he feels strongly about the name but also wants to do the right thing."
The Redskins were playing for the first time since President Barack Obama reignited the debate by saying he would "think about changing" the name if he were the team's owner. Congressman Tom Cole — a Republican — has called it "offensive."
The Oneida Indian Nation, which said it was running a radio advertisement in the Dallas market to protest the name Sunday, is also planning to meet with NFL representatives over the issue. The ad is part of a campaign slated to air in cities where the Redskins play this season.
A small group of protesters gathered near the stadium hours before kickoff, carrying signs that read "Stop Perpetuating Racism" and "Redskins = 81 years racism."
"I'm glad the president took a stand," said Yolanda Blue Horse of Dallas, a member of the Lakota Nation. "If Dan Snyder wanted to honour Native Americans, there's better ways to do that, rather than Redskins."
One of the protesters, Juan Mancias, drove from his home in the San Antonio area with flag of the American Indian Movement, which began in 1968. He said he protested a Cowboys-Redskins game at Texas Stadium in 1997.
Jack Flack, a Redskins fan from Washington who attended the game with friends, said he would change the name to Washington Senators.
"I mean, if the president's weighing in, it's probably time for change," Flack said.
Some Cowboys fans saw no reason for change.
"It's a tradition," said Stephanie Smith of Houston. "It's how it should be."
Cowboys owner Jerry Jones backed Snyder during the session with Goodell.
"It would be a real mistake to think that Dan ... in any way has a lack of sensitivity regarding somebody's feelings," Jones said. "I can promise you that."
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