President Donald Trump’s picks to run the Central intelligence Agency and the State Department likely will need votes from Democrats to get confirmed by the closely divided Senate, after a Republican lawmaker said he would oppose their nominations.

Sen. Rand Paul (R., Ky.) announced Wednesday he would oppose the nomination of Gina Haspel to lead the CIA, citing her involvement with the harsh interrogation program used in the wake of the Sept. 11, 2001, attacks. He also said he would oppose the nomination of Mike Pompeo to...