According to The Washington Post, the first couple had asked for Van Gogh's Landscape With Snow painting, featuring a man and his dog, to install in the White House.
But the Guggenheim's chief curator came up with a pointedly satirical counter-offer, emailing the White House Office of the Curator in September to say that the Trumps could instead borrow the toilet installation used by tens of thousands of visitors in a museum restroom.
Nancy Spector included a photograph in her email, saying: "We are sorry not to be able to accommodate your original request, but remain hopeful that this special offer may be of interest."
The 18-karat toilet, titled America, is Maurizio Cattelan's jab at the nation's greedier instincts.
It is fully functional and was used by more than 100,000 people, according to the museum, and has an estimated value topping $US1 million.
The toilet could be "a long-term loan", Ms Spector wrote.
"It is, of course, extremely valuable and somewhat fragile, but we would provide all the instructions for its installation and care."
The Post said the White House has stayed silent on the subject, while a Guggenheim spokeswoman declined to immediately comment.
Ms Spector has been openly critical of Trump on social media.
In a 2017 blog post she wrote that the "Trump reference" inherent in the gold toilet resonated with people "during the sculpture's time at the Guggenheim".
"When the sculpture came off view on September 15, Trump had been in office for 238 days, a term marked by scandal and defined by the deliberate rollback of countless civil liberties, in addition to climate-change denial that puts our planet in peril," Ms Spector wrote.
The President favours the colour gold, plating bathroom fixtures in his various homes with the precious metal and redecorating the Oval Office with gold-hued drapes.