The Chile forward, out of contract in the summer, has been a long-time target of the Premier League leaders City but they view the current asking price to acquire Sanchez in January as too great.
It is believed the cost quoted - including transfer, wages and agent fees - is in the region of £35million and City, who are hopeful an agreement can be struck for around £20million, are prepared to walk away from the deal if it does not come down.
They are also unconcerned if that opens the door for another club, with Manchester United boss Jose Mourinho having refused to dismiss possible interest from United.
City tried to sign Sanchez last summer and have been convinced that he wishes to work with his ex-Barcelona coach Pep Guardiola once more.
Though they had been willing to wait until Sanchez's deal runs out in June to try to land him on a free, they are under the impression he wants to move this month.
The knee injury Gabriel Jesus suffered on New Year's Eve had the potential to bring any transfer forward, yet the Brazilian could now be back training within three weeks.
Guardiola did not want to discuss the matter at his weekly press conference.
"I am not going (to speak about it)," he said. "I'm focused on (playing) Liverpool, believe me. Alexis - everyone knows how good he is, but I am not going to answer about Alexis or other players."