The Swans have been linked with a move for the young right-back, with several other teams also reportedly showing an interest.
Jenkinson's chances of an extended run in the Gunners' first team appear limited following the big-money arrivals of Mathieu Debuchy and Calum Chambers.
Chambers arrived from Southampton on Monday for a fee believed to be in the region of £16 million, while Arsene Wenger also splashed out around £11 million on France international Debuchy.
Jenkinson is said to be attracting the attention of the likes of Hull City, West Ham United and Sunderland, but Monk refused to be drawn on speculation linking him with a move for the talented youngster.
Speaking to Sky Sports News, the manager said: "We've been linked with every player under the sun! I think every club has. We've got our targets we're going to try and bring in and we'll keep that to ourselves.
"It doesn't always come off so you keep a plan B, but we know what we're doing as a club."
Monk also dismissed rumours that club-record signing Wilfried Bony will leave the Welsh side before the closing of the transfer window.
The Ivorian powerhouse had been linked with a big-money move to Liverpool, but Monk once again played down the transfer speculation when speaking to the South Wales Evening Post.
"It has to be made loud and clear what is happening at some point - but at the end of the day he is not for sale unless someone comes in with an astronomical offer, which is the same with every club around the world," he said.
"Unless anyone wants to pay an astronomical fee, he will be going nowhere. There will be interest in a number of our players - that's the way it is - but at the end of the day, Wilf is a Swansea City player," he said.
"Everything that has been said about him this summer has been said in the papers. We are not a paper company, so we'll just deal with the reality.
"At the moment nobody has come in for Wilf and he is our player, but it can't drag on for the whole summer. We don't want one of those scenarios where it's going on right up until the last day."
Swansea have already lost Ben Davies and Michel Vorm to Tottenham Hotspur this summer, while Spanish forward Michu has joined Napoli on a season-long loan.
However, Gylfi Sigurdsson has returned to the club, and the Iceland international was on target with two goals as they beat Plymouth Argyle 4-0 on Sunday.