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.

The family of Joseph Murphy has hit out at authorities for the delay.
Forensic tests have confirmed that the bullet was fired from a military-issue weapon.
Mr Murphy, 41, a father of 12, was injured in Ballymurphy, west Belfast, in August 1971 and died 13 days later in hospital.
Nine others were also killed during a series of shootings by paratroopers over three days.
The shootings occurred during Operation Demetrius, when paramilitary suspects were being detained and interned without trial.
The bullet was found in the body of Mr Murphy in October 2015.
Mr Murphy's family have said that they believe he was shot at a military barracks after being injured and that the bullet may still be lodged in his leg.
However, his daughter, Janet Donnelly, said that the delay in examining the bullet had hurt her family as they have been unable to rebury him.
"I received the report a couple of days ago," she said.
"They know rightly my daddy hasn't been reburied yet and won't be until all these tests are completed. And they're sitting on their hands doing nothing."
Mrs Donnelly's solicitor Mark O'Connor said: "This is significant because it confirms what we already suspected - that there was a bullet lodged in Mr Murphy's body which wasn't picked up by a post-mortem.
"This will help us to establish the type of weapon used and potentially the trajectory that it entered Mr Murphy's body."
Mr Murphy was shot in the thigh and taken to the Henry Taggart army barracks, where he said he was shot through the open wound a second time.
Earlier this year, Coroner Jim Kitson told a preliminary hearing an exhumation order had been granted because of the "exceptional circumstances" of the case.
Title favourites Roger Federer and Rafael Nadal will begin their US Open campaigns in New York on Tuesday.
[NEXT_CONCEPT]
A bullet found in the exhumed body of a man killed in shootings in Ballymurphy in 1971 was not sent for forensic examination for three months.