As Hillary Clinton prepares to make her choice for a vice-presidential candidate, Bill Clinton has privately expressed his support for Senator Tim Kaine of Virginia, according to three Democrats briefed on the conversations with the former president this week.

Mr. Clinton believes that Mr. Kaine, 58, a former governor and member of the Senate Armed Services Committee, has the domestic and national security résumé that both appeals to voters and makes him prepared for the presidency.

In an interview with Charlie Rose of PBS on Monday, Mrs. Clinton said she was “afflicted with the responsibility gene” and would value experience and preparedness over any other calculations in choosing a No. 2.

In recent days, Mrs. Clinton narrowed down her pick, and she is likely to appear with her running mate Saturday at a rally in Florida, though she could announce her choice in advance of the campaign event in a text to supporters.