WASHINGTON — For Representative Nancy Pelosi, Democrat of California, the issue of abortion rights has always been ancillary to her unwavering faith and deep approbation for generations of popes. “I actually agree with the pope on more issues than many Catholics who agree with him on one issue,” Ms. Pelosi said in an interview in her office at the Capitol last week.

But that one issue, abortion, is adding a thick layer of tension to the otherwise convivial mood as Congress prepares for the arrival of Pope Francis this week. The Capitol is ensnared in an imbroglio over funding for Planned Parenthood and a host of other abortion-related fights that could lead to a government shutdown next week.

Scores of House Republicans, responding to undercover videos claiming that Planned Parenthood affiliates illegally profit from selling aborted fetal tissue to researchers, are demanding that any bill to keep the government open be stripped of all federal funding for Planned Parenthood. The House passed two bills that would chip away at the organization and at abortion practices, and the Senate will bring its own abortion measure, perhaps the first of several, to the floor this week.