WASHINGTON—President Donald Trump said Thursday that an investigation of his campaign’s possible ties to Russia is a “witch hunt” and should be quickly brought to a close, but the appointment of a special counsel and recent grand jury subpoenas suggest a federal criminal probe is expanding to include other suspicious activity and countries.

“We look forward to getting this whole situation behind us,” said Mr. Trump, denying that he had urged James Comey, the recently fired director of the Federal Bureau of Investigation, to...