The Welsh worldwide had been close to agreeing to a new deal but Arsenal recently withdrew their offer amid input from head coach Unai Emery.
The Gunners midfielder is out of contract and free to leave the Emirates on a free transfer next summer.
Wenger signed Ramsey from Cardiff City for £4.8 million in 2008 and the Wales global has gone on to make 242 Premier League appearances for Arsenal, as well as twice scoring the victor in the FA Cup Final.
Speaking about Ramsey's contract situation at Arsenal, Wenger was quoted by Metro UK as saying: "We tried that before and it still can happen".
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But on this, his fourth visit to Pyongyang, he and Kim had lunch together before Pompeo left for South Korea . The Vatican's priests were expelled by North Korea long ago and state-appointed laymen officiate services.
Ramsey had the pick of the Premier league big boys when he moved to north London from Cardiff for around £5million in June 2008.
Liverpool had reportedly contacted Ramsey's agent to declare their interest but Dawson claims that the Reds have no interest in signing the Welsh worldwide. Ramsey, 27 will take his time to consider his options.
However, the report says Mourinho will urge Man Utd chiefs to do all they can to land Ramsey ahead of Liverpool as the Portuguese coach is a long-term admirer and feels the Welshman would be a great addition to his squad.
It would be a huge loss for Arsenal to let Ramsey go for free.