The 30-year-old remains a key player for the Gunners, but is being linked with a move away for the Emirates.
According to The Sun, Xhaka is wanted by Bayer Leverkusen, while Arsenal continue to be linked with both Declan Rice and Moises Caicedo.
The reports adds manager Mikel Arteta would not stand in the way of the player, who joined in 2016, but the club would seek £15million for his services.
talkSPORT host Woods hopes Xhaka remains, admitting she would be sad to see him go.
"I wouldn't want to lose him, Arsenal will be in the Champions League next season they'll need a bit more depth in that team.
"He's the unofficial captain in so many ways and is a bit of glue for some of those youngsters and new players coming in.
"I really wouldn't want to lose him. I'd understand if he wanted to have a move and play first team football consistently.
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"If you're bringing in a Declan Rice or Moises Caicedo and they'd edge him out. Then you wouldn't want to stop him doing that.
"I think he's still got more of his Arsenal career to live out. I adore him. He's been my favourite player since the whole revival of Granit Xhaka.
"I'd be gutted to see him leave."
talkSPORT's chief football correspondent Alex Crook added: "Granit Xhaka is an interesting one because I think when you look at this link at first to Bayer Leverkusen you would say 'well that can't happen'.
"To be fair to Granit Xhaka I think he's been just about the most improved player in the Premier League this season. He's been so impactful for Arsenal.
"Seven goals for Arsenal and he's getting assists as well, but he has only got a year left on his contract.
"We know Mikel Arteta is looking at revamping that midfield area. There's every chance they go back in for Moises Caicedo at Brighton having had a couple of bids for him turned down in January.
"We know Declan Rice is also a target.
"If a big money offer did come in, for a decent fee, for a player in the last year of his contract I think Arsenal could be tempted."