The Blues made a bid to sign Dumfries this past summer but the Dutch right-back remained at Inter with the Serie A side rejecting their proposal.
A new offer is being weighed up by Graham Potter's side which would see Dumfries sign for Chelsea in January before heading back to Inter on a six-month loan.
The Stamford Bridge side hold a good relationship with I Nerazzurri following the two sides' business in the summer transfer window.
Chelsea allowed club-record signing Romelu Lukaku to return to Inter on a season-long loan.
Meanwhile, Simone Inzaghi's side sold young Italian midfielder Cesare Casadei to the Blues for £13.5 million.
A move for Dumfries from Potter's side would see them potentially beat Manchester United to his signing.
The Red Devils were also interested in the right-back this past summer, with reports claiming Dumfries decided to stay at the San Siro for more game time.
There is competition for a right-back role at both Stamford Bridge and Old Trafford.
Manchester United currently possess Diogo Dalot and Aaron Wan-Bissaka as options for the right hand side of Ten Hag's defense.
Dalot has impressed at the start of the season with two assists in eight appearances.
Meanwhile, Chelsea have club-captain Cesar Azpilicueta and Reece James in their squad with the latter being touted as one of the best right-backs in Europe.
He has made seven appearances, scoring one goal and contributing an assist.
However, Dumfries offers a real attacking threat down the right.
His potential arrival could be as a replacement should either side be willing to part with one of their current right-backs.
Dumfries has made nine appearances this season, scoring two goals and grabbing an assist.
He has three years left on his current deal at the San Siro.
Chelsea could beat Manchester United to other Serie A player
Chelsea are reportedly the frontrunners in the race to sign AS Roma striker Tammy Abraham as they have a €80 million (£70.9 million) release clause in their former forward's contract.
Abraham has impressed during his time at the Stadio Olimpico, with 29 goals in 61 appearances, but a return to the Premier League may be on the cards.
Both the Blues and Manchester United both sought a striker signing this past summer.
Potter's side did manage to lure Pierre-Emerick Aubameyang to Stamford Bridge on deadline day.