The magazine, which exposed the alleged affair almost two weeks ago, brought forward its normal publication to make what it claimed were "new revelations".
Julie Gayet, the 41-year-old actress, was introduced to 59-year-old Hollande in 2011 by his former partner Segolene Royal, when he was a front-running presidential candidate.
It alleged that Trierweiler, 49, now resting at the presidential retreat La Lanterne in Versailles, wanted to remain President Hollande's partner and first lady, The Independent reported citing the Closer magazine report.
The magazine said Trierweiler has told her friends that she expects to accompany Hollande on a trip to Washington from Feb 9.
However, Elysee Palace sources have strongly hinted that Hollande plans to end their eight-year relationship.
Earlier, Hollande said he would "make a statement" before the Washington visit.
He is expected to reveal his intentions by early next week, before he comes to Britain for the annual Anglo-French summit with British Prime Minister David Cameron.