Arsenal are said to be keen on keeping Ramsey at the club until his contract expires at the end of the season.
But they may consider selling him now if the price is right – rather than losing him for nothing in the summer.
Juventus are understood to be dead set on landing Ramsey before another European powerhouse muscles in.
PSG, Bayern Munich and Real Madrid have all been linked with the Wales international.
But Juventus reckon they can get a deal done this month to block the advancement of their transfer rivals.
That is according to Italian newspaper Corriere dello Sport, who claim Juve director Fabio Paratici is ready to test Arsenal’s nerve.
Paratici is also looking to sign James Rodriguez from Real Madrid once his Bayern Munich loan spell ends in the summer.
But his focus right now appears to be on Ramsey and landing the 28-year-old before PSG get too close.
Corriere dello Sport claim Paratici wants to bring the player to Turin ‘immediately’ and they are willing to pay to get the deal done.
Another Italian outlet -Calciomercato – have claimed Juve have already lodged a £30.8 (€34m) offer for the midfielder.
Yet Juve appear ready to play the long game on Ramsey in order to land him for nothing in the summer.
They can agree a pre-contract deal with his representatives that effectively locks him down to a free transfer at the end of the season.
It is what they tried to push on to Emre Can while the midfielder was at Liverpool – but in the end Juve simply waited until the end of last season before signing the Germany international.
Having Can and Ramsey in midfield is the aim for Juve next term, with Rodriguez supposedly there to aid the attack spearheaded by Cristiano Ronaldo.
Juve are determined to maintain their dominance of Italian football and translate that on to the European stage.
Two Champions League finals in three years have given fans in Turin a taste of what is possible and Ronaldo was signed in the summer specifically to achieve that first European honour since 1996.
As for Arsenal, they are said to be monitoring a number of players who could replace Ramsey in Unai Emery’s midfield.