The British newspaper claim that the 25-year-old forward “has been made a top priority for the Magpies” in the summer transfer window.
Austin scored 18 goals and creating 23 chances in 35 Premier League appearances, had a pass accuracy of 66% and a shot accuracy of 67%, and won 86 headers, 15 dribbles and 16 tackles.
According to The Mirror, Rangers are “resigned” to losing the former Burnley star, but they will demand at least £15m for him.
Southampton, West Ham United and Aston Villa are also reported to be interested in the services of Austin.
Newcastle narrowly avoided relegation from the Premier League and do need a proven goalscorer.
The Magpies struggled for goals throughout the 2014-15 campaign, and Austin would be a suitable addition to the squad for next season.
However, whether or not Newcastle would be willing to pay £15m for the Englishman remains to be seen.