Xhaka is happy in London but is intrigued by the possibility of working under Jose Mourinho at Roma next season.
Mourinho has identified the Swiss midfielder as his top target and Roma are working to make sure that the Portuguese has Xhaka at his disposal when next season kicks off.
The Serie A giants tabled an initial offer of €15m for the midfielder, but Arsenal rate Xhaka at above that sum.
The north London club want a fee of €25m and, according to Italian outlet Calciomercato.com, the Giallorossi have gone close to meeting that figure.
It has been claimed that a second bid worth €20m has been tabled as Roma look to finalise an agreement.
The figure is still short of what Arsenal want but Roma are hopeful that an agreement can be reached soon.
Xhaka is keen on the move and has already agreed on personal terms on a four-year contract with Roma.
Roma have also proposed the idea of including a player in the agreement to sweeten the deal for Arsenal.