While football's world body denied any immediate closure, it estimated a loss of more than $30 million for 2016 and said the Fifa Council will hold talks on the museum's future at a January meeting.
According to the Fifa source, "the museum's new director told staff on Tuesday that Fifa has decided to close the museum."
The museum, which cost 140 million Swiss francs (130 million euros/$142 million), was opened in Zurich in February by Fifa President Gianni Infantino just after he took over from the disgraced Blatter. Infantino has since ordered widespread cost-cutting at Fifa.
The 3,000 square metre museum has had an average of 11,000 visitors a month, way below its target figure.
Signs of trouble emerged in October when director Stefan Jost left because of "differences of view" over the museum's strategy.
Fifa said in a statement that it has set up a task force led by deputy secretary general Zvonimir Boban "to develop a sustainable business model for the World Football Museum."
"In-depth analysis conducted as part of Fifa's on-going restructuring showed that the museum will generate an estimated loss of approximately 30 million Swiss francs (28 million euros/$30.4 million) in 2016.
"Its current operational concept would not allow it to do business viably beyond 2016."
The task force "will look into new, innovative models and concepts" including "a complete change of the museum's current physical set-up."
First proposals will be put to the body's ruling council in January, the statement added.
Fifa said it was in talks with Zurich city and staff.