The Croatian international laid on both of the Ibrox side's goals in their 2-1 win over rivals Celtic on Sunday - their first win on the Hoops' home patch since October 2010 - crossing for Ryan Kent to net the opener before his corner was headed home by defensive colleague Nikola Katic.
Now the former Osijek full-back is attracting attention, with Italian publication Calciomercato claiming interest from the Serie A duo.
Milan could offload Ricardo Rodriguez, who has played just five times this term, during next month's transfer window while Roma are rumoured to be looking for a deputy and long-term successor for 34-year-old left-back Aleksandar Kolarov.
Barisic has been capped 12 times for Croatia and despite an inauspicious start to life in Glasgow, has become a key part of Steven Gerrard's team. He has notched 13 assists from 27 games this term, scoring twice.