Page and Strzok both worked on the FBI investigation into Democrat Hillary Clinton's emails and, later, on special counsel Robert Mueller's Russian Federation investigation.
President Trump accused Page and her "FBI lover", Agent Peter Strzok, of "getting cold feet" about their testimony on Twitter Monday.
Congressional investigators were set for a private interview with Page on Wednesday, but her attorney said Tuesday she did not have time to prepare and would not appear.
In an interview with the Wall Street Journal earlier this year, Trump said the text between Strzok and Page amounted to "treason" and had exposed an Federal Bureau of Investigation plot against him. "In fact, Lisa and I went to the Federal Bureau of Investigation today to review the materials that were previously produced to Congress relating to her proposed interview, but after waiting for more than three hours, we were not provided with any documents".
If Page doesn't agree to either of these options, the chairmen of the two panels, Judiciary's Bob Goodlatte and Oversight's Trey Gowdy, told her lawyer that she would face contempt of Congress proceedings.
Lisa Page, pictured in January, was due to be interviewed by two House committees Wednesday. "She has an obligation to come testify". What are you hiding, Miss Page?
Rep. Devin Nunes (R-CA) threatened Page that he would enforce the subpoena, but has yet to do so.
"She could go to jail if they really want to press it for blowing off", Napolitano said.
House GOP lawmakers dispute Jeffress's exact timeline, arguing she had all the information she needed for an unclassified interview earlier in the day. "But a subpoena to testify before Congress is not optional; it's mandatory".
In response, House Judiciary Chairman Robert Goodlatte said Page has known for months about an interview, and the committee will use "all tools at our disposal" to obtain it.
In a statement released Wednesday, Amy Jeffress said Page requires more time.
But her lawyer said Page will testify at some point.
Strzok was questioned for 11 hours over his role and alleged bias in a closed hearing of the two committees two weeks ago.
In his tweet, Trump called Strzok and Page the "FBI lovers" in a reference to the fact that the pair were in a romantic relationship.