Not directly, no, but McConnell controls legislative calendar and see to it that the vote is taken.
That said, why would Mitch McConnell wait for "the coming weeks" for Franken to take his time about resigning? CNN is reporting that 32 Democrat Senators are calling for Al Franken to resign, so Franken's expulsion is a sure thing.
McConnell could have forced the expulsion vote on Franken last week; he doesn't have to wait for the outcome of the Alabama election. But McConnell didn't do it. He will wait until it best fits HIS needs, instead of controlling the media news cycle for Republicans.
And McConnell's need is to have leverage to force Moore out after he wins in Alabama. Therefore, McConnell will wait until then for Franken to resign, so he can demand a "bipartisan" purge in the Senate. If Franken went last week, the pressure would have been reduced on Moore after he wins. If McConnell ties Moore's expulsion vote to Franken's vote and schedules them for the same day, that will put enormous pressure on the other Republican Senators to vote for Moore's ouster after Franken's.
That's how Senate leadership can manipulate the ouster of a member.
-PJ
Moore will be allowed to present witness and specialist. Frankenstein waved that choice !