Representative Andy Ogles, a hard-right Tennessee Republican, on Thursday used Islamophobic language on social media to refer to Zohran Mamdani, the presumptive Democratic nominee for New York City mayor, and said he should be deported.

Representative Nancy Mace, Republican of South Carolina, implied that Mr. Mamdani was somehow tied to the Sept. 11, 2001, terrorist attacks, which occurred when he was 9. That came after Representative Marjorie Taylor Greene, Republican of Georgia, reacted on Wednesday to Mr. Mamdani’s apparent victory with an edited image of the Statue of Liberty clothed in a burqa.

The responses to Mr. Mamdani’s electoral triumph were the latest examples of how far-right Republicans in Congress have become overt in their use of bigoted language and ethnically offensive tropes, in both casual comments and official statements.

Mr. Mamdani, a three-term New York State assemblyman who is all but certain to win the Democratic primary for mayor, was born in Uganda and has lived in New York City since 1998, when he was 7 years old. He was naturalized as a U.S. citizen in 2018 and, if elected, would become the city’s first Muslim mayor.

There is no credible evidence to suggest Mr. Mamdani is not, or shouldn’t be, a U.S. citizen. But his shock win put him on the national radar, and some Republicans in Congress are now seeking to undermine him using a strategy similar to the racist one that Donald J. Trump employed against former President Barack Obama by questioning whether he was born in the United States.