However, a source close to Mustier said the 60-year-old Frenchman has no immediate plans to leave the Pegasus Europe special purpose acquisition (SPAC) vehicle he set up and invested in after leaving UniCredit a year ago.
The board of Switzerland's biggest bank is aiming to pick a new chairman by the end of the year and has included Mustier in a shortlist of possible candidates, said the sources, speaking on condition of anonymity as the matter is confidential.
UBS and representatives for Weber declined to comment.
The sources said Mustier may face investor criticism if he bows out before Amsterdam-listed Pegasus, which is focused on financial services, completes a merger.
"Timing is the main complication," one of the sources said.
Mustier, who teamed up with luxury tycoon Bernard Arnault and former Bank of America banker Diego De Giorgi to launch Pegasus in April, started his career within the corporate and investment banking division of Societe Generale.
He has previously been a contender for top jobs at some of Europe's biggest banks, including HSBC and Credit Suisse.
The board of UBS has been working with headhunters on a shortlist of possible candidates to succeed Weber for several months and will ask shareholders to approve its preferred candidate at its annual general meeting in April next year.
Weber will not seek to extend his tenure at UBS after his mandate ends in April, a source familiar with the matter said.
UBS is keen to enlist a chairman with an investment banking background who could work closely with former ING boss Ralph Hamers, who became CEO of the Swiss bank last year pledging to improve its digital services after a career focused primarily on retail banking.
One of the sources said Mustier's work ethic and discipline, combined with his investment banking expertise, would mean he could take an active approach and have a say in key matters.
"Mustier and Hamers have different but complementary backgrounds," one of the sources said. "Mustier has wholesale experience, while Hamers brings all the digital expertise."
UBS has been struggling to find qualified Swiss nationals for senior leadership roles and is open to the idea of offering the chairmanship to a non-Swiss citizen, the sources said.
Swiss banker Philipp Hildebrand, a former head of the Swiss central bank who joined BlackRock in 2012, has been ruled out from the shortlist, two of the sources said.
Jens Weidmann, a German known as a conservative policymaker who is due to leave his role as Bundesbank President on Dec. 31 for personal reasons, was initially seen as a possibility but he is subject to a one-year cooling off period once he leaves, making him an unlikely candidate for a transition in 2022.
Spokesmen for Hildebrand and Weidmann were not immediately available for comment.
Whoever succeeds Weber will take the reins of a bank with a healthy balance sheet.
In October UBS posted its highest quarterly profit over the past 11 years, with trading by the world's ultra rich fuelling a 23% fee income rise.
(Reporting by Pamela Barbaglia and Oliver Hirt; Additional reporting by Brenna Hughes Neghaiwi; Editing by Alexander Smith)
By Pamela Barbaglia and Oliver Hirt