SAWGRASS, Fla. (CNN) – It’s the dream pairing that golf fans have yearned for, but Phil Mickelson has upped the ante and called out Tiger Woods for a separate high-stakes shootout.

The veteran duo will play together, alongside another marquee player in 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 a challenge to his old rival.

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The pair, who have 19 majors between them, dueled at the top of the game for much of their careers, and are both enjoying a new flush — 42-year-old Woods after making a promising return from back fusion surgery, and Mickelson with a first victory in five years in Mexico earlier this season.

They were last in the same group four years ago. Mickelson sensed a chance to poke fun at the hype surrounding the pairing.

“I love that we’re paired together,” Mickelson, a five-time major winner, told reporters earnestly. “I think it’s really fun.”

Warming to his theme, the current world No. 19 added with a grin, “We haven’t been paired together in years, and as I look at the cover of the newspaper and the pairing is on there and the excitement that’s been going on around here, it gets me thinking, why don’t we just bypass all the ancillary stuff of a tournament and just go head-to-head and just have kind of a high-stake, winner-take-all match.”

“I don’t know if he wants a piece of me, but I just think it would be something that would be really fun for us to do, and I think there would be a lot of interest in it if we just went straight to the final round,” Mickelson added.

At a later news conference at Sawgrass, Woods was asked about Mickelson’s proposition. Pausing for effect before flashing a smile, the former world No. 1 — who has confirmed that he will play at the British Open in Carnoustie in July — said, “I’m definitely not against that. We’ll play for whatever makes him uncomfortable.”

Woods and Mickelson rarely played together on Thursdays and Fridays in their heyday, with tournament organizers keeping them apart to spread their top talent across the TV schedules. They were never friends as they battled for supremacy, but they have become closer in recent years through Woods’ role as a vice captain in the Ryder Cup and Presidents Cup, and, in turn, Mickelson’s support for his old foe during his many back issues.

Tiger Woods and Phil Mickelson talk on the 11th hole during a practice round prior to the start of the 2018 Masters Tournament at Augusta National Golf Club on April 3, 2018 in Augusta, Georgia. (credit: Andrew Redington/Getty Images)

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Their surprise practice round together at Augusta ahead of The Masters set tongues wagging about golf’s new bromance.

“We’ve always had a mutual respect over the years, and I’ve always appreciated what he’s done for the game of golf,” added Mickelson. “Fifteen years ago, my record against him sucked, and now it’s OK. I’m doing better as time has gone on.”

Mickelson said Woods’ performances in the early 2000s — he won four straight majors from 2000-2001, including the U.S. Open by a record 15 strokes — was the “benchmark” and the “greatest” golf ever played. “I don’t think anybody… will ever see that level of play again,” Mickelson said.

“It was the most remarkable golf in the history of the game, and I think unrepeatable. I think it was that good. I look at 2000 as being kind of the benchmark, at the U.S. Open as being the greatest golf I’ve ever witnessed and I believe ever has been played,” Mickelson said. “And it sucked to have to play against him. It really did.”

Mickelson added: “The guys today look back, and they say, come on, how much better could he have been and so forth, and it just goes to show you that they weren’t there to witness it.”

For his part, 14-time major winner Woods said that Mickelson was the name he always looked for on a leaderboard. “We have always looked at each other and said, where is he on the board,” he said. “That’s what Arnold [Palmer] and Jack [Nicklaus] used to do all the time. They’d always try and find, okay, what’s Jack at. And the same thing with Jack, where’s Arnie at. It’s been either way for our entire careers.”

“For me to be able to play with a person I’ve gone against for over two decades, it’s a lot of fun,” Woods said.

The Players Championship, held over the exacting Stadium course with its iconic island-green 17th hole, attracts the deepest field of the season and ranks for many as akin to a major. It is often dubbed golf’s “fifth major.” Jordan Spieth even said that it was “harder to win than a major.”

“If you win here, you can win anywhere else,” Spieth told reporters.

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