Federal prosecutors are weighing bringing child-pornography charges against former congressman Anthony Weiner over sexually explicit exchanges he allegedly had with a 15-year-old girl, according to people familiar with the matter.

Mr. Weiner, a New York Democrat, is being investigated by the U.S. attorney’s office in Manhattan and the Federal Bureau of Investigation, which seized his electronic devices, including a laptop and a cellphone, as part of the probe.

Officials initiated the investigation last fall, after the Daily Mail in the U.K. reported that Mr. Weiner had exchanged sexually explicit messages and photos with the girl.

In recent weeks, according to some of the people familiar with the matter, attorneys for Mr. Weiner have had discussions with federal prosecutors in Manhattan in hopes of dissuading them from bringing charges, or at least from bringing the most serious one: production of child pornography, which carries a 15-year mandatory minimum prison sentence upon conviction.

These types of discussions can indicate both sides are trying to reach an agreement in which the defendant pleads guilty in exchange for a less-severe charge.

Mr. Weiner could face the production charge, some of the people familiar with the case say, because he allegedly solicited explicit images from the teenager. Prosecutors also are weighing other charges, including receipt of child pornography, for which conviction carries a five-year mandatory minimum, and possession of child pornography, which has no mandatory minimum.

Prosecutors could decide not to bring any charges.

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