Gurlitt, who died in May aged 81, had secretly stored hundreds of works by the likes of Chagall and Picasso at his Munich apartment and a house in nearby Salzburg, Austria.
The collection, worth an estimated €1 billion ($1.26 billion), contains an as yet undetermined number of works taken by the Nazis from their Jewish owners during World War II.
The Berne Art Museum will accept the bequest, but only pieces for which restitution claims can be ruled out will come to Berne, the Sonntagszeitung reported.
"According to well informed sources, the meeting of the museum's board of trustees on November 26 will just sign off on the already detailed agreement," the paper wrote.
A spokeswoman for the museum said it was still in talks with Germany and the German state of Bavaria, and that "current speculation" about the collection was to a significant extent inaccurate.
"The talks are proceeding constructively, but are not yet concluded," she said. "In light of this, it is unnecessary to comment on the speculation, which is in significant parts incorrect."