The former Chelsea protege has long been linked with a return to the Premier League since making a name for himself in Italy, winning the Europa Conference League last year.
Abraham has amassed 36 goals and 12 assists from 98 games in a Roma shirt over the past two seasons, including nine goals and seven assists from 45 outings in the current campaign.
The 25-year-old has come under some criticism this term for his patchy form, but he has responded with two goals in Roma's last three Serie A games, including the opener in Saturday's draw with reigning champions AC Milan.
In the fourth minute of second-half injury time, Abraham advanced into the box and picked out the bottom corner from Zeki Celik's pass, although Alexis Saelemaekers came up with a dramatic late leveller three minutes later.
According to Gazzetta dello Sport, representatives from Man United and PSG were in attendance to keep an eye on Abraham, and Roma are braced for bids for the attacker this summer.
However, Jose Mourinho's team have Abraham under contract for another three years, and Chelsea also have first refusal on any transfer for their ex-academy product with an €80m (£70.1m) buyback clause.
While Chelsea are expected to be in the market for a new number nine this summer, it is unclear if the Blues have any desire to reunite with Abraham, who has refused to rule out a return to Stamford Bridge.
In any case, Chelsea will be notified of any bids that come in for Abraham, and sporting director Tiago Pinto will hold out for the full value of his release clause from interested parties.
The report adds that Tottenham Hotspur scouts were also in attendance at Stadio Olimpico, although it is unknown if the North London club were specifically keeping tabs on Abraham.
A permanent successor to Cristiano Ronaldo will be Man United's number one priority this summer, as Wout Weghorst's loan spell from Burnley is not expected to be made permanent, while the injury-plagued Anthony Martial could be on the way out too.
Tottenham talisman Harry Kane is still believed to be the Red Devils' top attacking target, but they will not be held to ransom by Daniel Levy, who has no intention of selling to a direct Premier League rival.
Napoli's Victor Osimhen and Eintracht Frankfurt's Randal Kolo Muani have been mentioned as potential alternatives, but the former would cost in excess of £100m, while Frankfurt intend to keep hold of Kolo Muani for at least one more season.
Meanwhile, Lionel Messi appears to be inching closer to an exit from PSG, who have supposedly decided against renewing the seven-time Ballon d'Or winner's contract beyond the summer amid talk of a sensational Barcelona homecoming.
Kylian Mbappe's long-term future remains a hot-button topic too, while Mauro Icardi and Reims loanee Hugo Ekitike - who has struggled to adapt to life at the Parc des Princes - could be on the chopping block this summer.