The newspaper said three people with knowledge of the couple's situation said Weiner, 46, and his 35-year-old wife Huma Abedin are expecting their first child. The couple, who have been married less than a year, have disclosed the pregnancy, now in the early stages, to close friends and family, the Times said.
Abedin, an aide to Secretary of State Hillary Clinton, has not made any public comment. She departed Wednesday for a trip to Northern Africa with her boss.
U.S. Rep. Allyson Schwartz of Pennsylvania Wednesday became the first Democratic House member to call for the New York congressman to resign.
Schwartz was followed by Rep. Michael Michaud, D-Maine, who agreed Weiner should step aside, CBS News reported.
Speaking with reporters at the White House Wednesday, Obama administration press secretary Jay Carney said, "We have no comment on that story."
Weiner has found himself scandalized by revelations he sent risque pictures of himself to several women via the Internet and had inappropriate phone conversations of a sexual nature with women, none of whom he says he ever met in person.
CBS reported Schwartz, who has the task of recruiting candidates to run for Congress next year, said in a prepared statement she found that behavior enough to show him the door.
"Having the respect of your constituents is fundamental for a member of Congress," she said. "In light of Anthony Weiner's offensive behavior online, he should resign."
The network said one aide to a member of New York's congressional delegation acknowledged "momentum is building for Weiner to go, but he seems to be dug in."
The Times also reported a group of conservatives tracked Weiner on Twitter, sending warnings to young women he followed.
The group, #bornfreecrew, was led by a man named Dan Wolfe, at least on Twitter, with the handle @PatriotUSA76, the newspaper said Wednesday. Wolfe apparently never gave his real name, and his account is no longer on Twitter.
One tweet Wolfe sent April 14 showed how he was tracking the congressman: "Weiner's new follow is a high school girl. LMAO! Freak!"
Weiner admitted at a news conference this week he sent inappropriate pictures of himself to young women. He has denied knowingly sending them to underage girls or using government equipment to do so.
Members of both parties have been calling for his resignation. Tim Kaine, a former Virginia governor and Democratic National Committee chairman, said Weiner should resign, WCAV-TV, Charlotteville, N.C., said.
"Lying is unforgivable, public lying about something like this is unforgivable, he should resign," Kaine said.
Reince Priebus, head of the Republican National Committee, accused Democrats of hypocrisy Wednesday during a media breakfast in Washington, Politico reported.
He pointed out Rep. Chris Lee, R-N.Y., quit quickly after news that he had sent a shirtless photo of himself to a woman he met on Craigslist became public.
"If these folks were exhibiting a little leadership here in this three-ring circus that's come about, they should tell him that he should hit the bricks," Priebus said.
2014: The Year in Music [PHOTOS]