WASHINGTON—Senate Majority Leader Mitch McConnell (R., Ky.) said he has enough votes to set impeachment trial rules that don’t guarantee new witnesses would be called, rebuffing demands from Democrats.

Mr. McConnell has concluded he has at least 51 votes from within his 53-member Republican conference for opening the trial first, and then voting later on whether to issue subpoenas for witnesses and documents. Mr. McConnell has insisted that a trial take place in two parts, with opening statements first, followed by votes on…

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