In the interviews, Conway "impermissibly mixed official government business with political views about candidates in the Alabama special election," the independent federal ethics agency said in a report submitted to Trump for "appropriate disciplinary action."
The Hatch Act prohibits federal employees from using their official capacity to affect or interfere with elections, although some other political activities are allowed.
The office said it gave Conway a chance to respond to its allegations but she did not.
White House spokesman Hogan Gidley disputed the report, saying in a statement that "Conway did not advocate for or against the election of any particular candidate.
"She simply expressed the president's obvious position that he have people in the House and Senate who support his agenda," Gidley said.
In the two TV appearances, Conway discussed the contest between Democratic candidate Doug Jones and Republican Roy Moore for the seat vacated when Jeff Sessions became Trump's attorney general.
In a Fox News Channel appearance on Nov. 20, Conway talked about why voters should not back Jones, and in a CNN interview on Dec. 6 she laid out why they should support Moore, the report said. Jones won the election and is now in the Senate.
"The President must take swift and serious disciplinary action against Ms. Conway," Representative Elijah Cummings, the top Democrat on the House of Representatives Oversight Committee, said in a statement. "Anything else sets a terrible example."
Walter Shaub, the former head of U.S. government ethics watchdog the Office of Government Ethics, filed a complaint after Conway's first appearance, saying it violated the 1939 law.
Conway, who served as Trump's campaign manager before taking on her role as White House senior counselor, has previously run afoul of Shaub and other presidential ethics experts who criticized her for publicly endorsing the clothing and jewelry line sold by Trump's daughter and White House adviser Ivanka Trump.
The White House later said her praise for the products was inadvertent, and no disciplinary action was taken.
(Reporting by Susan Heavey and Ayesha Rascoe; Editing by Jeffrey Benkoe and Dan Grebler)