WASHINGTON — President Obama on Tuesday will call attention to what he has said is an “embarrassment” in America: the fact that women make, on average, only 77 cents for every dollar that a man earns.

But critics of the administration are eager to turn the tables and note that Mr. Obama’s White House fares only slightly better. A study released in January showed that female White House staff members make on average 88 cents for every dollar a male staff member earns.

The dueling statistics reveal the political sensitivities around a set of gender-related issues that could be critical in the midterm elections this fall. Those include pay equity, family leave, preschool and child care.

Mr. Obama and his Democratic allies are trying to portray Republicans as insensitive to the concerns of women, in the hopes of capitalizing on the kind of lopsided female support that helped Mr. Obama win the White House in 2008 and 2012. On Tuesday, Mr. Obama is to sign an executive order barring federal contractors from penalizing employees who discuss their compensation.