Ivanka Trump has largely cast herself as a behind-the-scenes force for moderation in the West Wing, but this week she will take her first real political risk when her signature parental leave plan is presented to a conservative House majority hostile to any new government mandates.

Her proposal for a $25 billion federal paid leave program, part of the budget plan that President Trump will release Tuesday, is a reflection of Ms. Trump’s influence in her father’s inner circle. The plan has been her primary area of interest since the 2016 campaign, when she pushed her father to introduce a version of it on the stump.

While she has become a popular figure with some Republicans on Capitol Hill, she will discover how far her sway extends. As she tries to leave her first tangible imprint on the government with a plan that cuts against Republican orthodoxy, a buzz saw will surely await. The question is just how big it will be.

“If it is going to be a buzz saw, I hope it’s going to be an intelligent buzz saw,” said Representative Peter T. King, Republican of New York. “I think we have to realize that the ground has shifted. I don’t mean the political ground, I mean the family ground.”