The Swiss Football Association said that current Austria coach was their top choice.
"The ball's in Marcel Koller's court," SFA boss Peter Gillieron said.
Koller's two-year contract with Austria is up for renewal on December 31st and he is currently considering his options.
He will not be able to play the two neighbours off against each other, however.
"There's a certain amount of time-pressure because the Austrian Football Association wants him to give an answer by November 4th," they said, without elaborating.
"But the SFA won't be changing anything in the contractual conditions on offer."
Former midfielder Koller, 53 next month, made his Switzerland debut in 1982 and was capped 55 times before hanging up his boots after Euro 1996.
Moving into management, he won the Swiss title in 2000 with Saint Gallen and again in 2003 with his hometown club Grasshoppers Zurich, before mixed fortunes in the German Bundesliga with Cologne and Bochum.
In 2011 he was tapped to lead Austria.
They failed to qualify for Euro 2012, finishing third in their group behind Turkey and top-ranked Germany.
In the race to reach the 2014 World Cup in Brazil, they were also third, behind playoff contenders Sweden and, yet again, table-topping Germany.
Hitzfeld announced on October 17 that he would be retiring from the game after the World Cup, at the age of 65.
The former Borussia Dortmund and Bayern Munich boss has been at Switzerland's helm since 2008 and has succeeded in taking them to their third World Cup in a row.
Unbeaten in the qualifiers — albeit in a relatively easy group along with Iceland, Slovenia, Norway, Albania and Cyprus — they jumped seven notches to seventh in the FIFA rankings and are among the top seeds for December's finals draw.
The SFA chose Hitzfeld to succeed iconic local name Köbi Kuhn, in charge from 2001.
Kuhn, who took Switzerland to Euro 2004 and the 2006 World Cup, had already decided to retire after Euro 2008 on home turf.
Under Hitzfeld, Switzerland qualified the 2010 World Cup, beating future champions Spain in their opening game but failing to clear the group stage.
His star fell sharply after their failed Euro 2012 qualifying campaign, but he developed new talent and thanks to the 2014 World Cup run has earned the nickname "Gottmar" — a play on the German for "God".