WASHINGTON—President Trump drew a direct connection between the special counsel investigation into alleged Russian interference in the 2016 election and his decision to revoke the security clearance of former CIA Director John Brennan and review the clearances of several other former officials.

In an interview Wednesday, Mr. Trump cited Mr. Brennan as among those he held responsible for the investigation, which also is looking into whether there was collusion between the Trump campaign and the Kremlin. Mr. Trump has denied...