Edin Dzeko started the season with a sensational goal and Roma could reward him with a new and improved contract.
The Bosnia international scored the late winning volley to beat Torino 1-0 on Sunday in the Serie A opener.
According to La Repubblica newspaper, the 32-year-old is being offered a new deal beyond his current contract to June 2020.
It would include a pay rise to around €5m per season.
Dzeko had been the subject of a strong offer from Chelsea in January, but both club and player turned it down and he went on to reach the Champions League semi-finals.
He first arrived at Roma from Manchester City in 2015, on a €4m loan with €11m option to buy, and has since scored 74 goals in 130 competitive games for the Giallorossi.
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