The 30-year-old, who arrived in Turin from Arsenal in 2018, has struggled for game time so far this season under Massimiliano Allegri, with the Serie A opener against Udinese being his only league start of the term.
Having arrived on a free transfer from Arsenal, the Old Lady are said to be willing to let Ramsey leave for free in January to free up a section of the wage budget, and Everton and West Ham United have already been linked with a swoop for the Wales international.
According to The Sun, Newcastle have now entered the race for the 30-year-old but could be put off by his large wage demands.
Midfielder Joe Willock was the only piece of incoming business for the Magpies in the summer transfer window, and manager Steve Bruce has emphasised the importance of strengthening his squad, with the St. James' Park outfit yet to pick up a league victory this season.
Newcastle have collected just three points from the first six games of the new Premier League campaign and sit level on points with Leeds United in the relegation zone as a result.