FDR and some of his staff, especially his Press Secretary Steve Early, reacted to Eleanor Roosevelt's activism for civil rights with frustration and sometimes anger. While Mrs. Roosevelt didn't oppose segregation, she didn't openly support it, and began getting involved in pressuring her husband to pass legislation like an anti-lynching bill.
At that time, however, the South was called "The Solid South" by Democrats, because it always voted for Democrats. FDR counted on the southern vote to help keep him in the White House, so he couldn't risk alienating segregationists in Congress. He tried to get his wife to quiet down and stop pushing him and others, but it didn't work very well.
Here's a link to a great discussion about Eleanor Roosevelt's work on race issues.