Woods also won the 2006 British Open at Royal Liverpool.
World No 9 Day is making his first appearance since the US Open last month where he battled hard in the final group on Sunday and finished tied ninth despite suffering from vertigo.
Now on medication to control a viral infection in his right ear, the cause of the vertigo, Day is confident there won't be a repeat of his physical collapse from Chambers Bay's second round.
The trio will head out at 9.55am local time in the 20th game of the 144th Open.
Former world No 1 Adam Scott will be on the other side of the draw, heading out at 2.12pm local time in game 41 with German two-time major winner Martin Kaymer and American Jimmy Walker.
World No 2 Jordan Spieth, fresh off his fourth win of the year at the John Deere Classic, looks to keep the grand slam alive when he goes off at 9.33am.
The Masters and US Open champion Spieth has been grouped with the man he conquered at Chambers Bay last month in Dustin Johnson and Japan's Hideki Matsuyama.
Australian Rod Pampling will be part of the opening game in the tournament, grouped with Dane Thomas Bjorn and England's Greg Owen at 6.32am.
- AAP