The veteran duo will play together, alongside another marquee player Rickie Fowler, in the first two rounds of the Players Championship at Sawgrass, Florida, this week, and 47-year-old Mickelson was in playful mood as he threw out the challenge to his old rival.
Woods found himself afforded cut-line hope as Justin Thomas, one of golf's most unflappable figures, fluffed a chip at the 18th en route to a bogey.
That's been a familiar theme since he finished tied for 32nd at the Masters.
He did well to two-putt for par at 17 from the rear fringe after watching both his playing partners find the hazard, but he then tugged a long-iron into the hazard from the tee at the last. The last win for Woods at The Open came 12 years ago at Royal Liverpool, when he finished 18-under par on the weekend. I didn't hit the ball close enough. "I feel like I'm putting well".
Woods said he welcomed the idea, suggesting they play for whatever "makes him uncomfortable". "It was a bad combo".
A crucial eagle on the ninth hole changed his round, however, and could end up being the score that gets Woods back in the tournament. Woods wasn't a whole lot better. He doesn't have his best stuff obviously.
The third member of the previous champions trio Rickie Fowler may exclusively shoot 74, whereas one other big-name triumvirate to misfire was that of Rory McIlroy (71), Jordan Spieth (75) and Justin Thomas (73).
Woods nearly had a bonus when a long-range putt lipped out at the eighth, and his par at the long ninth rounded off another day of frustration for the former world No 1, who insisted his biggest fault was with his approach play.
"I just didn't want to drop a shot, didn't want to shoot an over-par round", Woods said. "He's feeling better about his health. I think they might have tipped back a couple and got a little sleepy". "His chances of winning a Major are maybe better than winning a regular event".
Speaking to Independent.ie, championship director Simon Alliss admitted the addition of Woods to the line-up would take to "another level" while also boosting ticket sales.
A closer looks reveals Woods hit just 12 greens in regulation and was never inside 10 feet with his approach shots.
"But to be honest, I cared more about having a good routine and hitting a good shot in this golf tournament than the record". In last week's Wells Fargo Championship, he barely made a putt over 3 feet. On Friday, the only place he gained strokes was off the tee.
"I said it Sunday at Wells Fargo, I was anxious about energy this week", Mickelson said. "I wish I would have been a little more committed to what I was seeing with that grain [on the greens]". The cutline kept flipping between 1-under and 2-under all day.
Woods failed to make the cut at a tournament earlier this year, and is looking at yet another disappointing finish here.
"If I had one round with a PGA Tour golfer, it would probably be Tiger Woods". "I didn't get off to the start I wanted to". Everyone was just trying to keep it together.
Tiger's Troops should brace themselves.