Strzok was removed this summer after the inspector general of Justice Department discovered him and another person from Mueller's team exchanging anti-Trump text messages.
Days later, then-FBI Director James Comey announced that he would not be recommending charges against Clinton for mishandling classified information on her private email server.
A spokesperson for the special counsel's office, Peter Carr, said "Immediately upon learning of the allegations, the special counsel's office removed Strzok from the investigation", reported New York Times.
The president's most vociferous defenders in Congress have called for a special counsel to investigate how the Federal Bureau of Investigation handled the Clinton probe, and other Clinton-connected matters.
The Post has repeatedly sought comment from Strzok and Page, but got no response.
The inquiry into Mr. Strzok is being conducted by the Justice Department's inspector general, Michael E. Horowitz, who is leading a broad examination of how the F.B.I. handled the Clinton email investigation.More news: Florida Gators Rumors: Dan Mullen 'Targeted' to Be Gators' Next Coach
Justice Department spokeswoman Sara Isgur Flores said in an email that "we disagree with the chairman's characterization and will continue to work with congressional committees to provide the information they request consistent with our national security responsibilities".
Current and former law enforcement officials who worked with Strzok - who remains assigned to the FBI's human resources department - told the Times that they are unaware of any incidents in which the agent allowed his political views to influence investigations.
It is not clear what Mr. Strzok said in his text messages.
Mueller is overseeing the FBI's investigation into Russian meddling in the 2016 presidential election and possible coordination with the Trump campaign.
On Saturday, following the reports about Strzok, the inspector general's office said in a statement that as part of its investigation, "the OIG has been reviewing allegations involving communications between certain individuals, and will report its findings regarding those allegations promptly upon completion of the review of them".
However, the President later backed down.