The Wales international is in the final year of his contract, and according to Steve Stammers in the Mail on Sunday, while the Gunners are said to be confident he will stay put, they want his future resolved before the transfer window closes because they otherwise risk losing him for nothing next year.
Chelsea are keeping a keen eye on the situation and are prepared to make a late move for him should the two parties fail to reach an agreement.
Per Sami Mokbel for the Daily Mail, Arsenal have offered Ramsey a five-year deal to remain at the Emirates Stadium, which would see the midfielder receive a "substantial pay increase" on the £110,000 per week he earns.
New boss Unai Emery is said to be eager to retain Ramsey and has plans to make him a key part of his side.
Encouragingly for Chelsea, though, is that it seems the 27-year-old will be allowed to leave this summer if he does not sign a new deal, per Football.London's Charles Watts:
It is of little surprise Arsenal are keen to tie him down. The midfielder was one of the Gunners' top performers last year, and that was recognised by the Emirates faithful:
With 11 goals and 12 assists to his name in 32 appearances in all competitions, last season was one of his most productive in an Arsenal shirt.
His versatility makes him an important asset too, and he performed virtually every role in midfield at one time or another under former manager Arsene Wenger.
Few players seemed to notably improve under Wenger during his final years at the club, but with Emery in charge it's possible Ramsey and others can take their games to new heights.
The Welshman recently gave an update on the negotiations:
He could be an astute buy for Chelsea, particularly for a fee in the region of £30 million in today's transfer market.
It will put Arsenal in a difficult position, though, as they could be left with the choice of losing him for free or letting him join their London rivals.