Wales Online reports that Swans boss Paul Clement is pushing for a move to bring the 20-year-old midfielder to the Liberty Stadium, with Bayern in favour of loaning him for the season to gain experience.
Clement worked with Sanches during his time as assistant to Carlo Ancelotti at Bayern and could persuade the midfielder to reunite with him in south Wales.
The former Real Madrid assistant coach said over the weekend he was looking to bring in additional attacking options before the transfer window closes on 31 August.
"I had my phone on right before kick-off against Palace," Clement was quoted as saying by Wales Online.
"And once it goes back on I'm hoping we're going to get some positive news in the next days about bringing in one or two quality players who are going to add to this group."
Sanches joined Bayern from Benfica for €35m (£32m) last summer but has failed to establish himself at the Allianz Arena.
He has not played for the Bavarian giants since the German Super Cup win over Borussia Dortmund on 5 August and was left out of the club's squad for the 2-0 win over Werder Bremen at the weekend.
Ancelotti said last week that he was open to letting Sanches leave on loan for the season.
"If he stays I am satisfied. If he wants to leave it's okay as well. No problem," the Italian was quoted as saying by the Sun.
Swansea sold Gylfi Sigurdsson to Everton for £45m on 16 August and brought in Hull City's Sam Clucas to replace the Iceland midfielder.