Tim Kaine’s dual jobs, Democratic senator from Virginia and vice-presidential candidate, collided on Tuesday during a visit to Florida, when he called on Republicans to bring the Senate back into session to deal with the Zika threat given the outbreak in the state’s south.

“Congress should not be in recess when Zika is advancing,” Mr. Kaine told a crowd in Daytona Beach, urging quick approval of $1.1 billion in health funding that has been the subject of a fierce and lengthy partisan fight.

That brought a sharp retort from Don Stewart, a spokesman for Senator Mitch McConnell, the Kentucky Republican and majority leader.

“Oh, he’s going to drop his filibuster?” Mr. Stewart asked, referring to the refusal by Senate Democrats to take up a House-passed Zika measure that includes what they see as “poison pills,” including restrictions on use of the money by Planned Parenthood.