According to Darren Lewis in the Mirror, following last summer's interest in Alex Oxlade-Chamberlain and January's purchase of Olivier Giroud, Chelsea are set to return to the Emirates Stadium with a view to bolstering their squad by moving for Ramsey.
The Blues are preparing to overhaul their squad ahead of next season after a poor defence of their Premier League title, which sees them sitting fifth in the top flight.
Ramsey's contract with Arsenal runs until June 2019, so he could be available for a relatively cheap fee if the Gunners fail to tie him to a new deal by the end of the season.
Per BT Sport (via Lewis), the Wales international recently said there has been little progress on new terms: "Nothing to comment on at the moment. I think they've been busy to tie up other contracts at the moment."
The 27-year-old is enjoying a fine season despite Arsenal's numerous woes during the 2017-18 campaign. He has netted seven goals and provided seven assists in a combined 21 appearances in the Premier League and UEFA Europa League.
Chelsea's midfield is in a state of flux.
N'Golo Kante remains a potent force—although he has not been quite as impressive this season as he was while winning the league with Chelsea and Leicester City in the previous two campaigns.
However, summer signings Danny Drinkwater and Tiemoue Bakayoko have both disappointed at Stamford Bridge, while Cesc Fabregas looks to be on the wane.
Ramsey is reaching his peak and has the Premier League experience that should mean he fits in easily at Chelsea.
If Arsenal fail to get him signed up to a new contract before the summer transfer window opens, it could be a sensible move from the Blues to bring Ramsey to Stamford Bridge.