Putin and Trump are expected to meet in Finland's capital of Helsinki on July 16 to discuss bilateral ties and various issues on the international agenda. The summit will be their first full-fledged meeting since Trump's taking office in January 2017.
Trump's daughter Ivanka and his son-in-law Jared Kushner will not accompany the US leader, the media outlet added.
Shine, former co-president of Fox News and Fox Business Network, joined the Trump administration earlier in the month. Shine, who was at Fox News for some 20 years, resigned from the broadcaster in May 2017 over sexual harassment allegations.