Federer has been drawn in the same half as six-time champion Novak Djokovic, who is only seeded 14th after his lengthy absence with elbow problems and was handed a tricky draw.
Should he reach the fourth round, a clash with fourth seed Alexander Zverev looms, although the young German may well have to beat his brother Mischa to get there.
Anderson and Edmund played each other at the French Open last year, when the South African prevailed over five tough sets.
Anderson said: "It was a very close match, five long sets. He's a terrific player, he's really improved, very physical, his forehand's improved so it's going to be a tough match."
In the women's draw, world No.1 Simona Halep faces exciting 17-year-old Australian Destanee Aiava in round one, with Eugenie Bouchard a potential round two opponent and Petra Kvitova a possible foe in round three.
Konta, seeded ninth, has reached the semifinals and quarterfinals of the tournament in the last two years and is Britain's big hope following the withdrawal of Andy Murray.
Second seed Caroline Wozniacki meets Mihaela Buzarnescu of Romania in a much more inviting section of the draw but Maria Sharapova, back at the scene of her failed drugs test two years ago, has a lot of work to do.
After German Tatjana Maria in round one, Sharapova could face 14th seed Anastasija Sevastova, who ended her US Open run, while in-form former champion Angelique Kerber is a potential round three opponent.