The Ecuador international has featured regularly for Swans boss Paul Clement during pre-season and still has three years left on his contract.
However, the club are said to be open to offers having become frustrated with the winger's injury problems.
The 27-year-old managed just 14 appearances last season due to a variety of injuries, prompting Swansea to seemingly welcome enquiries as to his availability.
Newcastle United have previously been linked with Montero, while Al-Rayyan, who are managed by former Swans boss Michael Laudrup, have made contact with the Welsh outfit.
There are thought to have been no officials bids for the winger yet, despite reports of an offer of £3million.
Clement appears to be keen to overhaul his options out wide having last week sanctioned Mo Barrow's move to Reading, although that deal has since stalled after the Royals had an offer rejected.