The 24-year-old defender has impressed during his three months at Hillsborough, despite The Owls languishing at the foot of Division One, and he seems happy in South Yorkshire.
Monk is due to return to St Mary`s this week, but he wants to stay on and aid Wednesday as they battle to avoid relegation.
He said: "I`ve enjoyed it here and my priority is first-team football.
"Hopefully over the next couple of days, we`ll sit down and have talks with Gordon Strachan. If things are sorted out, I`d be happy to move.
"There are nine more games and we can still survive. You`re not down until the maths say you are."
