The 22-year-old has impressed with his displays at the World Cup so far this summer, attracting the attention of top European clubs.
According to reputable Italian journalist Gianluca Di Marzio, Juventus have offered €19m, plus add-ons, for the CSKA Moscow star.
However, his club have rejected the deal and have raised their asking price from €25m to €30m following Golovin's impressive World Cup displays.
It remains to be seen whether Juventus will renew their offer for the player, or whether they will turn their attentions elsewhere.
Since the start of the 2017-18 season, Golovin has recorded eight goals in 47 appearances for club and country - a good return for a young midfield player. Those figures include one goal and two assists so far at the World Cup in his home country.
Despite his young age, Golovin already has experience of playing in the Champions League and Europa League, so he would not have to adapt too much if he were to make the move to Juventus.
Juventus have been heavily tipped to sign English midfielder Jack Wilshere, following his decision to leave Arsenal after a 17-year spell at the north London club. However, Goal now report that the Serie A champions do not have former England international Wilshere on their radar as they look to boost their midfield options.