The Swiss climber died in Nepal on Sunday during an acclimatization trek for summiting Everest by a new route.
According to Swiss tabloid Blick, both the president of Steck's home village of Riggenberg and the president of Grindelwald are enthusiastic about the idea of honouring the climber by naming a spot on the Eiger after him.
“We want to take up and implement this idea,” said Grindelwald's Christian Anderegg. “Ueli Steck has more than earned this honour”.
Steck, nicknamed the ‘Swiss machine', first climbed the Eiger aged 18 at the start of an incredible climbing career that took him all over the world.
Among his many achievements, Steck climbed Everest without oxygen and ascended the south face of Annapurna solo.
In 2015 he succeeded in climbing all 82 Alpine peaks over 4,000m in 62 days.
Later that year he set the solo speed record for ascending the Eiger north face for the third time, reaching the summit in two hours, 22 minutes and 50 seconds and taking back the record from compatriot Dani Arnold.
Arnold had broken Steck's previous record in 2011.
“The Eiger was his favourite mountain and he set many records here,” Anderegg told Blick.
A memorial plaque could be mounted on the Hireli ridge section of the Eiger, he said, with one local suggesting the spot be renamed ‘Ueli-Steck-Hireli'.
Any name change would have to be approved by the cantonal and federal authorities, said the paper.