Top Senate Republicans called Thursday for a new special counsel to probe how the FBI and Justice Department handled interactions with the Trump campaign and the investigation into Russian meddling in the 2016 election.

Sens. Charles E. Grassley of Iowa and Lindsey Graham of South Carolina, the Republicans who’ve been pursuing the issue, said an inspector general’s investigation is already under way, but it’s constrained by law and it will take someone with special-counsel powers to get to the bottom of what happened.

“Thus, we believe that a special counsel is needed to work with the Inspector General to independently gather the facts and make prosecutorial decisions, if any are merited,” said Mr. Grassley and Mr. Graham, who were joined in their request by Sens. John Cornyn of Texas and Thom Tillis of North Carolina.

The top Republicans on the key committees in the House already called for a special counsel, and the senators’ letter ups the pressure on Attorney General Jeff Sessions and Deputy Attorney General Rod Rosenstein.

The new probe would be in addition to special counsel Robert Mueller, who is looking into Russian meddling in the 2016 presidential election and whether the Trump campaign colluded with Russian figures.

Republicans said the new probe would look at the decision to appoint Mr. Mueller, as well as look at Justice Department and FBI decisions in 2016 and 2017 about snooping on a Trump campaign figure under the Foreign Intelligence Surveillance Act.

The application for the FISA warrant to conduct surveillance relied in part on a controversial dossier drawn up by British ex-spy Christopher Steele, and funded by the Democratic National Committee on behalf of Hillary Clinton’s campaign.

The senators said the FISA application is at odds with Mr. Steele’s own claims in a British court about his findings.

They demanded that if Mr. Sessions and Mr. Rosenstein didn’t name a special counsel, the two men provide “a detailed reply explaining why not.”

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