The Italian side would expect to take the Ukraine striker on a free transfer.
The 32-year-old has a year left on his contract but has been told he can leave Stamford Bridge and are even considering paying up his £7million a year contract rather than welcoming him back after a loan spell at AC Milan.
Rare highlight: Andriy Shevchenko scores for AC Milan against Fiorentina in March
Milan have little interest in making a renewed move and new Chelsea boss Carlo Ancelotti is understood to be not keen on a player he used sparingly at the San Siro last season.
Roma, who have little money to spend on new players, see Shevchenko as attractive - according to Italian newspaper Corriere dello Sport.
Roma could lose Montenegro striker Mirko Vucinic this summer, with Manchester United mentioned as a possible destination, making their need to add attacking options even more pressing.
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