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The FSB, the G20's regulatory task force, is charged with making sure bank regulations are strong enough to prevent another financial crisis.
Heading the FSB would not require the incumbent to give up his central bank governorship. An announcement is expected at the November 3-4 G20 summit.
Hildebrand has been a vocal proponent of tougher bank regulation after the Swiss government had to bail out flagship bank UBS
Hildebrand and Carney are good friends and studied together at Oxford.
The Swiss government is quietly backing Hildebrand's candidature for the FSB, though some right-wing politicians oppose it, the paper also said.
"Of course the federal council supports SNB Chairman Philipp Hildebrand's candidature for the top job at the FSB," Finance Minister Eveline-Widmer Schlumpf told the SonntagsZeitung.
But the government was doing this via diplomatic channels and not publicly, the paper also quoted her as saying.
Hildebrand has come into the crosshairs of the SVP before.
Party mastermind Christoph Blocher has called on Hildebrand to resign after the SNB posted its biggest annual loss ever last year due to currency interventions to weaken the Swiss franc.
(Reporting by Catherine Bosley; Editing by Helen Massy-Beresford)