WASHINGTON—Attorney General Jeff Sessions said Thursday he would remove himself from involvement in any investigation related to the 2016 presidential race, following the disclosure that he had conversations with a Russian official while advising the Trump campaign.

Lawmakers from both parties had called on Mr. Sessions to recuse himself after reports that he met with the Russian ambassador to the U.S. last year, even though he had testified in Senate confirmation hearings that he had no contact with Russian officials during...