In a letter posted on Twitter, Mr Schumer wrote: "Given the growing popularity of FaceApp and these national security and privacy concerns, I ask that the FBI assess whether the personal data uploaded by millions of Americans on to FaceApp may be finding its way into the hands of the Russian government, or entities with ties to the Russian government."
The app allows people to submit a photo of themselves and uses artificial intelligence to transform their face to look older or younger, as well as changing the colour of their features.
FaceApp has said said it does not sell or share data with third parties.
The UK's data watchdog has said it is "considering" concerns raised about the FaceApp service over potential privacy issues.
The Information Commissioner's Office (ICO) did not say whether it was investigating the app but warned people to be wary about what details they share when downloading any app.
"We are aware of stories raising concerns about FaceApp and will be considering them," a spokeswoman said.
"We would advise people signing up to any app to check what will happen to their personal information and not to provide any personal details until they are clear about how they will be used."
One of the concerns highlighted by users on social media was how FaceApp is able to use photos created on the service, though others have pointed out that some of these terms are not dissimilar to those on many other platforms.
American lawyer Elizabeth Potts Weinstein described it as "a licence to use your photos, your name, your username, and your likeness for any purpose including commercial purposes".
In response to concerns, FaceApp chief executive Yaroslav Goncharov said it does not sell or share any user data with any third parties.
"FaceApp performs most of the photo processing in the cloud," he said.
"We only upload a photo selected by a user for editing.
"We never transfer any other images from the phone to the cloud.
"We might store an uploaded photo in the cloud.
"The main reason for that is performance and traffic: we want to make sure that the user doesn't upload the photo repeatedly for every edit operation.
"Most images are deleted from our servers within 48 hours from the upload date."
Mr Goncharov also said that the company accepts requests from users to remove all their data from its servers.