The Jazz hope the veteran small forward will bring leadership and toughness to their young lineup.
A source, who spoke on condition of anonymity because no deal has been finalized, told The Associated Press Howard is hoping to decide where to sign early this week.
The Jazz made adding an athletic shooter their top priority in free agency after re-signing backup point guard Earl Watson on Friday.
"We've had serious discussions," Jazz general manager Kevin O'Connor acknowledged Saturday when asked about Howard.
At the time, O'Connor said the team had not made a formal offer to Howard, who is from North Carolina.
Many teams want to see Howard work out because he is coming back from a serious knee injury.
Howard was a key player for Dallas from 2003-10 and helped the Mavs in their run to the NBA Finals in 2006. He averaged a career-high 19.9 points in 2007-08.
Howard was traded to Washington in February 2010 but tore his ACL the following month.
He only appeared in 18 games for Washington last season and averaged 8.4 points
"I remember his Dallas days before he tore his ACL," Jazz forward-center Al Jefferson said. "He was a special player. It takes a couple of years to bounce back and be 100 percent from an ACL because I lived it. But I know Josh coming here or anywhere he goes he's trying to prove and redeem himself, show everybody he's still got it. If he (does) come here, it will be a plus for us."
They also re-signed Watson to a two-year deal as backup to Devin Harris.
The 6-3 Tinsley is an eight-year NBA veteran who most recently played in the NBA's developmental league in Los Angeles.
He had off-court issues in Indianapolis but said he is hungry to get back in the league and do things right.
The Jazz historically have carried three point guards and it may be more important now because of the condensed season.
McLeod has played four seasons in the NBA with four different teams, including the Jazz from 2004-2006. He averaged 6.6 points, 3.3 assists and 1.6 rebounds in 22 minutes for the Jazz before being traded to Golden State after the 2006 season.
Information from ESPN The Magazine's Chris Broussard and The Associated Press contributed to this report.