No one was hurt during the Friday night incident.
"Mr. Jones was here just trying to be a normal customer, trying to bowl" with several others, Franklin police detective Stephanie Cisco said, according to The Tennessean newspaper.
Clayton Smith, 33, instigated the confrontation at the Franklin Family Entertainment Center when the dispute broke out, Cisco told the paper. Smith brandished a small pocketknife and "threatened to beat up Mr. Jones and to use the knife on him," Cisco said.
Smith was led from the bowling alley and arrested. According to The Tennessean, he has a history of arrests that includes charges of stalking, unlawful weapon possession and drug offenses, Cisco said.
Titans coach Jeff Fisher did not return a message from the newspaper.
Fisher, who talked with Jones by phone Thursday night, doesn't expect his cornerback to return to the Titans' facility with the rest of his teammates when they report March 19 for offseason conditioning.
The NFL is examining 10 separate incidents in which Jones has been questioned by police since being drafted, including a Feb. 19 triple shooting at a Las Vegas strip club.
Fisher and chief operating officer Steve Underwood met with Jones' lawyers at the attorneys' request Friday for what Fisher characterized as an information-gathering meeting.
Fisher said the Titans are still gathering their own information before deciding whether to keep Jones.
"As an organization, it's always been our policy to gather facts and to make sure we have all the information, and that's what we're doing," he said. "At this point, I don't see him coming back on the 19th."
Fisher was clearly upset that Jones failed to inform team officials about two arrests in Fayette County, Ga., last year, including one related to a search for drugs at the home Jones bought for his mother.
"Typically speaking, if there's a situation like this, it's the player's responsibility to inform the club, and that did not take place," Fisher told reporters Friday.
Jones' attorneys, Worrick Robinson and Manny Arora, said on radio station WGFX on Friday morning that they didn't expect Jones to be charged in the Las Vegas episode but acknowledged that their client too often is in the wrong place at the wrong time.
"If he doesn't commit to changing it, at some point it's going to be too much," Arora said. "It may already be there to some extent."
Information from The Associated Press was used in this report.