A spokesman for Hillary Clinton said the allegations are a distraction from the probe into Russian collusion in the 2016 election.
Peter Schweizer claims that Clinton wouldn't meet with Kazakh officials until concessions were granted to a Clinton Foundation donor.
Shep Smith calls statements others on the network have made about the conspiracy theory "inaccurate."
Accusations between Democrats and Republicans come a week after special counsel Robert Mueller moved forward with charges against Trump campaign members.
Republican lawmaker says Mueller "must step down immediately” because of ties to the Obama administration.