The Premier League leaders feel the overall price for signing the Chile forward this month - transfer fee, agent costs and wage demands - is now too high and must come down.
Recent reports indicate that City's January move for Sanchez may also fall apart, though, and arch-rivals United are keen to jump in on the move.
However, Woods revealed there is a third player United are considering letting move to the Emirates Stadium this month: "Mourinho might be able to offer Arsenal an incentive in any deal by allowing centre-back Chris Smalling to go to north London".
The prospect of Sanchez going to United in the manner that Robin van Persie once did seems completely unconscionable to some Arsenal supporters.
But we shouldn't rule out the possibility of Sanchez - and Arsenal, who you imagine would be reluctant to sell him for nothing in the summer - eventually accepting the offer tabled by City, now playing a game of "who blinks first" with the Gunners.
"I don't think it's ethical or correct to be speaking about players of other clubs", Mourinho said during his pre-match press conference on Friday ahead of Monday's Premier League home clash with Stoke City.
The young Brazilian would be an exciting signing, for sure, but it's a sad indictment of where Arsenal are that two of their main rivals argue over their star player as the Gunners take a punt on a kid as a replacement.
"I think it's a good investment because we are talking about a top player and I'm hearing that the amount is around £20m. The only word I can say is the same that everybody says - he's a phenomenal player". We don't believe in signing a player just to do something.
Guardiola said his focus was on maintaining City's fine form when they travel to Liverpool in the Premier League on Sunday. There is always a way to find a balance.
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