WASHINGTON—President Donald Trump and his legal team intensified attacks on the special counsel’s probe into possible Russian election interference, departing from a previously more cooperative posture and prompting more urgent warnings from senators not to undermine the investigation.

In his first Twitter post to target the special counsel, Robert Mueller, by name, Mr. Trump on Saturday said the probe “should never have been started in that there was no collusion and there was no crime.” On Sunday, the president mentioned...