It is the first time the two have been face to face since Garcia made a "fried chicken" comment directed at Woods on May 21 at a European Tour function outside of London.
Garcia has since apologized, both in a statement and during a news conference, and Woods said at the Memorial Tournament two weeks ago that the matter was done, that it was time to move on.
Still, there had been no face-to-face apology from Garcia. Asked about their brief exchange after a practice round Monday, Woods politely said "I'm not going there" and simply made reference to a Tuesday news conference in advance of the U.S. Open at Merion.
The driving range where the handshake took place is on Merion's West Course, where the players practice this week before shuttling to the East Course, site of the tournament.
The tension heightened between the two golfers last month during the third round of the Players Championship, when Garcia suggested Woods showed poor form in pulling a club from his bag -- thereby causing spectators to cheer -- just as he was about to hit a shot, one that he hit poorly.
From there, a series of jabs went back and forth between Garcia and Woods in the media, with gallery marshals offering different takes on the matter, and neither player seemingly willing to budge and move on.
But their ire escalated when Garcia's attempt at humor went bad. He was jokingly asked if he would be having dinner with Woods this week during the U.S. Open and replied: "We'll have him 'round every night. We will serve fried chicken."
Such references are viewed as a racial stereotype, and Woods did not take kindly to the remarks, saying on Twitter that they were "wrong, hurtful and inappropriate."
Garcia, while still at the BMW Championship three weeks ago, told reporters he wished to speak to Woods.
"We talked to his manager and asked him if he wanted us to call Tiger, or wait for Merion and do it there face to face," Garcia said. "And they said they would rather do it there. There's nothing we can do, so we'll wait until we get there and we'll talk."
Garcia, who is also due to speak to media on Tuesday, apparently made the first effort on Monday. He then played several practice holes accompanied by a police officer.