There have been murmurs of some sort of exhibition between the two golfers going back to the Players Championship in May, when they were grouped for the first two rounds and Mickelson suggested they skip "the ancillary stuff of a tournament and just go head-to-head and just have kind of a high-stakes, winner-take-all match."
When presented later with the idea, Woods didn't shoot it down.
"I'm definitely not against that," he said then. "We'll play for whatever makes him uncomfortable."
Woods' agent, Mark Steinberg, declined to comment Friday on the report. Mickelson's agent, Steve Loy, could not be reached.
But Mickelson, who is playing at the Greenbrier tournament this week, told Golf.com that the original plan was to play the match in Las Vegas this past Tuesday, but all of the various television, marketing and financial details could not be worked out in time.
"We're working on a different date," Mickelson said. "I thought it was done for the third, but obviously it wasn't."
Although both players have amassed fortunes, Mickelson said the $10 million amount is enough to get their attention.
"It's a ridiculous amount of money," Mickelson said. "No matter how much money you have, this amount will take both of us out of our comfort zone."
Woods, 42, and Mickelson, 48, have had a long rivalry that dates to their junior golf days in Southern California. At times, their relationship has been frosty; the 2004 Ryder Cup when they were paired together -- and lost both matches -- often is cited as an example.
But in recent years, their relationship has mellowed. They've become more friends than foes, thanks in part to their work together as part of the U.S. Ryder Cup effort. Following Mickelson's controversial comments at the 2014 Ryder Cup in Scotland, where he criticized the PGA of America and captain Tom Watson, big changes were made.
As part of those, a Ryder Cup committee evolved, and its two permanent members are Woods and Mickelson.
Also unknown, until Woods disclosed it, was that Mickelson had reached out to offer help when Woods was struggling with his back issues.
"Phil and I have a great banter," Woods said at the Players Championship in May. "We give each other the needle, we always have. But I think our relationship has certainly gotten a lot closer with me being a vice captain the last couple teams and sitting there and having very lengthy conversations with him about things, not just the pairings but just things in general.
"When I was trying to deal with the nerve [pain] in my back and trying to come back and play and I wasn't very good, he always texted me some very encouraging words. And then on top of that, when you guys all saw how I was chipping so poorly, my nerve and my back was not going very good, and I was flinching a lot, and he offered numerous times to help me out with technique and just talk about philosophy."
Mickelson told Golf.com that he hopes the match leads to more exhibitions between the two golfers. And while there is a reported $10 million on the line to the winner, it is more than likely that both players will walk away with a hefty sum due to the selling of sponsorships, marketing and television rights.
As for official golf, after the Greenbrier, Mickelson is scheduled to play a practice round Monday at the site of the Ryder Cup in France as well as a practice round at Carnoustie, site of The Open in two weeks. Mickelson is playing next week's Scottish Open, and Woods' next start will be The Open.