Franken says he is resigning after accusations by several women of sexual misconduct. That sets up a November 2018 election to serve the final two years of his term.
Republicans have also identified former Sen. Norm Coleman as a possible candidate.
Coleman has already posted on his personal Facebook page that he won't run. He narrowly lost to Franken in 2008 in a close election that included a monthslong vote count.
Pawlenty ran for president in 2012.
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Minnesota Republicans are reaching out to former Gov. Tim Pawlenty about running in a race to replace Sen. Al Franken.