The Republicans spoke on condition of anonymity to describe a decision they'd been told privately.
Several men in US politics, media and entertainment have been accused of sexual harassment and misconduct in the last few months.
Earlier, liberal Democrat Senator Al Franken, of Minnesota, announced his resignation after facing allegations of sexual harassment by at least eight women. Franken said some of those accusations were false and said he remembered others differently than his accusers did. He said he'd depart in a few weeks.
Rep. John Conyers, Democrat, Michigan, resigned effective immediately. He also faced accusations from women of improper sexual behaviour that he's contesting.
Franks drew a sharp response from Democrats during a 2013 House committee debate when he said "the incidence of rape resulting in pregnancy are very low". He sought to clarify the comment, saying later-term abortions linked to pregnancies caused by rape are infrequent.
He's a strong backer of President Donald Trump and has embraced some of his stances on social issues. Franks has harshly criticised some NFL players for not standing during the national anthem, calling them "arrogant and overpaid Lilliputians who dishonour America".
Franks represents a district encompassing suburbs north and west of Phoenix. He serves on the House Judiciary and Armed Services committees.
Before winning election to Congress, he served in the Arizona legislature and founded the Arizona Family Research Institute, an organisation associated with Dr James Dobson's "Focus on the Family". The institute advocates for policies designed to protect children and families.
- AP