Strzok was exchanging the texts with Lisa Page, an Federal Bureau of Investigation lawyer with whom he was having an extramarital affair, The Post reported.
President Trump drew criticism for his apparent acceptance of Russian President Vladimir Putin's denials of election meddling.
The Times reported "Strzok and a colleague reacted to news events, like presidential debates, in ways that could appear critical of Mr. Trump".
A lawyer for Mr. Strzok declined to comment.
The Justice Department inspector general's office said in a statement Saturday that its investigators are "reviewing allegations involving communications between certain individuals, and will report its findings regarding those allegations promptly upon completion of the review of them". Page was also on Mueller's team but left two weeks before the alleged anti-Trump texts were found. NBC News has not confirmed the contents of the messages and other details of Strzok's removal. He was reassigned last summer to the FBI's human resources department after the Justice Department's inspector general began looking into the text messages, the papers said, quoting several unidentified people familiar with the matter.
Page also no longer works much with McCabe, these people said.
It is not clear what Mr. Strzok said in his text messages.
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A friend of Page said there was no professional misconduct at issue in the matter, and she said Page left the Mueller team in July before any of the issues came to light.
"Immediately upon learning of the allegations, the Special Counsel's Office removed Peter Strzok from the investigation", Special Counsel's Office spokesperson Peter Carr told BuzzFeed News. Word of the texts could give new fuel to those demands.
Trump has criticized the FBI's handling of the Clinton email investigation, initially citing it as his reason for firing former FBI director James Comey on May 9. Days later, Comey announced the emails on the laptop did not change the FBI's view of the case. Mr. Trump seriously entertained the idea but ultimately backed down.
Mr Flynn pleaded guilty on Friday to lying to the FBI and agreed to co-operate with Robert Mueller's investigation.
During the Clinton email investigation, Page was a regular participant when Comey would hold "skinny group" meetings on the case - a small collection of advisers that gathered to address sensitive cases, according to people familiar with the case.
A former senior Trump administration official said Strzok was even-handed in all of his dealing with the Trump White House.