INDIANAPOLIS—The NFL and its players are on the verge of agreeing to a new, decade-long collective bargaining agreement after a tense sequence of events over the past week that left the billionaires who run the league anxious and the millionaires who play inside it preparing for the possibility of a labor war.

But those fears were allayed after a four-hour meeting here when NFL owners met with player leadership in a room that included the likes of superstars Aaron Rodgers and Richard Sherman along with influential owners…

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