Franks, 60, a member of the conservative House Freedom Caucus, represents Arizona's 8th Congressional District, a mainly suburban area of Phoenix.
The Associated Press, CNN and Politico also reported Thursday afternoon that Franks will resign, citing multiple unnamed sources.
Franks, a self-styled "conservative, Reagan Republican", joined Congress in the early 2000s and has hewed conservative on a range of issues in his time there, with a particular focus on anti-abortion issues. "And if this turns out to be true, there won't be that many people who are surprised".
Conyers denied the allegations, while Franken said some of the accusations against him were untrue and he remembered other incidents differently from his accusers.
"Half of all black children are aborted", Franks infamously said in 2010.
It was not immediately clear why Franks was resigning. Three years later, Franks pushed bills that would limit access to abortion, even in cases of rape or incest, saying "the incidents of rape resulting in pregnancy are very low". But no revelations were aired in the aftermath of the decision, and Franks went on to assume the chairmanship of a House Judiciary subcommittee.