According to The Athletic, the Italian club general manager Tiago Pinto is set to arrive in London over the next 48 hours to seal the signing of the 23-year-old striker.
The report adds that Roma have already agreed on a deal with Chelsea.
There have been reports that Roma will sign the player on loan with an obligation to buy but Athletic claim that the Italian side are prepared to purchase him outright this summer.
Chelsea have inserted a buy-back clause in the player’s contract.
The Italian Serie A outfit are now looking to close the deal for the England international by the end of this week.
Abraham needs to leave Stamford Bridge in order to get regular game time, and a move to Roma could be ideal for him.
He is likely to lead the line for the Italian club next season, and he should get ample time to showcase his potential on the pitch.
The 23-year-old has managed to score a total of 30 goals across the last two seasons for Chelsea, and he has the potential to develop into a top-class forward for Roma in the coming years.