The striker spent the past season on loan at West Bromwich Albion.
Despite impressing on an individual basis, he did not do enough to help his team gain promotion to the Premier League.
The Chronicle says Stoke City have shown a serious interest in acquiring Gayle.
But it will take a figure of nearly £20 million to tempt Newcastle into a sale.
With the future of Salomon Rondon uncertain, Gayle could be the main striker at Newcastle at the start of the Premier League season.