By Andrew Beaton  Updated May 30, 2018 1:45 p.m. ET

President Donald Trump didn’t mince words last fall when he explained to Dallas Cowboys owner Jerry Jones that he wouldn’t relent in his criticism of NFL players who were kneeling during the national anthem to protest social injustice.

“This is a very winning, strong issue for me,” Mr. Trump said in a phone call, according to a sworn deposition given by Mr. Jones and reviewed by The Wall Street Journal. “Tell everybody, you can’t win this one. This one lifts me.”

Mr. Jones was deposed in a grievance filed against the National Football League by former San Francisco 49ers quarterback Colin Kaepernick, who contends that National Football League teams have blackballed him over his anthem protests.

A White House official said that Mr. Trump was advising Mr. Jones on what he believed would be good for the country and good for the sport. “The majority of the American people agree with the president, love our country, love our flag and believe it should be respected,” the official said.

Nearly two years since Mr. Kaepernick’s initial protest, NFL owners last week voted to change league rules: Players on the field for the national anthem are required to stand, or their teams could face repercussions. The overhaul allows players to remain in the locker room for the anthem, which was previously banned, but it also permits teams to punish players that violate the new protocol.

“I brought it out,” Mr. Trump said of the issue in a Fox & Friends interview after the rule change was announced. “I think the people pushed it forward.”

Depositions given by Mr. Jones and other owners indicate that Mr. Trump’s criticism pushed the league to shift its stance. League executives publicly repeated the NFL’s aim to stay out of politics. But privately, they made political calculations in response to Mr. Trump’s repeated hammering of the issue.

The controversy over anthem protests had already been raging for a year when Mr. Trump—at a stump speech in Alabama last September—said that if a hypothetical player knelt during the national anthem, his team’s owner should “get that son of a bitch off the field now.” On Twitter , he later encouraged owners to fire those players and suggested a boycott.

“I was totally supportive of [the players] until Trump made his statement,” Stephen Ross, the Miami Dolphins’ owner and creator of programs advocating for social justice, said in his deposition. Noting that owners’ conversations with Mr. Trump were relayed during a league meeting, he said: “I thought he changed the dialogue.”

Mr. Trump’s stance is a key point in Mr. Kaepernick’s grievance, which was filed last October. It alleges that the league and its 32 teams colluded to keep him unsigned last season because of his political views.

Mr. Kaepernick, who ignited the anthem demonstrations in 2016 to draw attention to racial inequality and other social justice issues, has remained unsigned despite statistics superior to other quarterbacks who have gotten jobs. His grievance argues that Mr. Trump was an “organizing force in the collusion” because of the president’s relationships with various NFL owners, many of whom have backed him with campaign contributions.

When the 2017 season began, only a handful of players were still kneeling. But Mr. Trump’s fiery comments in Alabama—just before the season’s third weekend—changed that. The following Sunday, players knelt en masse to directly rebuke the president.

Many owners took a knee alongside their players. Mr. Jones, in a high-profile Monday night game Sept. 25, knelt with his players before the anthem—but they stood when it was played.

Publicly, the NFL fought back and touted the moment as a display of unity. Commissioner Roger Goodell called Mr. Trump’s comments “divisive.” The league’s chief spokesman, Joe Lockhart, called the president “out of touch” and said, “everyone should know, including the president, that this is what real locker-room talk is.”

Behind the scenes, the kerfuffle rankled a league that was already grappling with declining ratings. Messrs. Ross, Jones and Bob McNair, the owner of the Houston Texans, said in their depositions that they believed the protests were financially hurting their teams.

Some owners were upset with the comments made by Mr. Lockhart, a former press secretary for President Bill Clinton, who they believed was furthering the feud with the president. Mr. Lockhart, who declined to comment, left the league after the season.

“You cannot piss off a large percentage of your constituency,” Mr. Jones said in his deposition. Regarding Mr. Lockhart, he said: “I was proud to see him go.”

Mr. Jones relayed his conversation with Mr. Trump in a meeting among owners to decide how to handle these protests, according to Mr. Ross’s testimony. Many owners disagreed with the president and his tactics. Mr. McNair, the Texans owner, said he didn’t like the players kneeling, but he thought Mr. Trump’s language was inappropriate. “I wished he hadn’t said it,” Mr. McNair said in his deposition. Representatives for Mr. McNair didn’t respond to requests for comment.

After Mr. Trump’s comments, Mr. Ross met with various Dolphins players several times and asked them to stay off the field in lieu of protesting. Later, “they informed the coach that they couldn’t, in their conscience, stay in the tunnel. They wanted to go out.” Miami’s coach allowed them. And they did.

Mr. Ross said Robert Kraft, the owner of the New England Patriots, also brought up a conversation with Mr. Trump with the owners. Mr. Kraft told the group he was friends with Mr. Trump, but thought the president was wrong in the way he handled this issue.

Mr. Goodell, the NFL’s commissioner who at the time was at the center of a budding war among the owners over his contract extension, was also influenced by Mr. Trump’s comments, Mr. Ross said in his deposition.

“We continue to abide by the confidentiality provision of the [collective bargaining agreement] and will not comment on the grievance,” an NFL spokesman said.

This off-season brought both new opportunity and scrutiny. The league’s owners had two scheduled meetings, in March and May, to discuss an array of topics, including the anthem. At the same time, a former teammate of Mr. Kaepernick’s who also had taken a knee during the anthem, Eric Reid, was going unsigned.

In May, Mr. Reid filed a collusion grievance against the league, like Mr. Kaepernick. The NFL Players Association also filed a grievance, saying a team violated league rules by asking Mr. Reid about his intentions during the anthem.

Then, when the owners met last week in Atlanta, the host city for next year’s Super Bowl, they changed the rule.

Mr. Jones declined to comment. In his deposition, which was taken before the rule change, he fought back against the idea that Mr. Trump reframed the conversation.

“Let’s [not] give him that much credit,” he said. “But I recognize he’s the president of the United States.”

—Louise Radnofsky contributed to this article.