The pair have never met at Flushing Meadows but could face each other in the semi-finals this year.
"I'd be happy to play him here - I don't think we are thinking that far ahead," said Federer.
Women's world number one Karolina Pliskova, defending champion Angelique Kerber and Britain's Aljaz Bedene are also in action on day two.
Czech Pliskova opens play on Arthur Ashe Stadium at 16:00 BST against Poland's Magda Linette, followed by Germany's Kerber against Naomi Osaka of Japan.
"When I came here one week ago and I stepped on the site, a lot of great memories came back," said Kerber.
"It's a great feeling for sure to be back here at the US Open, for me it's a really special place."
Top seed Nadal begins against Serbia's Dusan Lajovic in the third match on Ashe at about 20:00 BST.
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Third seed Federer faces American teenager Frances Tiafoe in the second night match at about 02:00 on Tuesday, after American Madison Keys plays Belgian Elise Mertens.
"It's good to be number one of the world again, of course," said Nadal, who replaced Andy Murray at the top of the rankings earlier this month.
"I think it's a good achievement after three years having some troubles. So I'm just happy for that, excited to start the US Open."
Bedene plays Andrey Rublev of Russia in the fourth match on court 11 at approximately 23:00.
Title favourites Roger Federer and Rafael Nadal will begin their US Open campaigns in New York on Tuesday.