CBS says the interview is being taped Thursday in Los Angeles.
Last week, Johnston filed papers to run for office in his hometown of Wasilla, Alaska, next year.
While Johnston has not officially declared which political office he aspires to, his manager, Tank Jones, has said Johnston is interested in a run for mayor or City Council. Earlier this month, Jones confirmed the 20-year-old Johnston planned to run for office as part of a reality TV show.
Wasilla is where Palin, the former Alaska governor and the Republican vice presidential candidate in 2008, got her political start, first as a council member and then as mayor.