President Donald Trump on Wednesday renewed his public pressure on Attorney General Jeff Sessions, suggesting the Justice Department should be looking at the Obama administration as it probes possible ties between the Trump campaign and Russia.

In a tweet Wednesday, Mr. Trump wrote: “If all of the Russian meddling took place during the Obama Administration, right up to January 20th, why aren’t they the subject of the investigation? Why didn’t Obama do something about the meddling? Why aren’t Dem crimes under investigation?...