The Zurich-based bank is believed to be the latest to bend to Washington's demands to release names of staff who could have helped clients avoid paying tax in the United States, Le Temps newspaper said.
In addition to providing personal information about staff to the US tax authorities, Baer is said to have made available personal documents, emails and details of telephone calls.
The data concerns current and former staff, as well as outside contractors, the report added.
It cites as its source lawyer Douglas Hornung, the legal representative of a former HSBC executive whose name was one of more than 1,000 given to the US tax office.
Referring to "three different sources" in his open letter to Swiss president Eveline Widmer-Schlumpf, Hornung writes that Bär "gave more than 2,500 dossiers to the US, and that HSBC had handed over approximately 1,100", violating Switzerland's federal law on data protection.
That brings the total number of names handed over by the 11 banks in Washington's sights to 10,000, Hornung claims.
President Widmer-Schlumpf, who is also the country's finance minister, recently announced that she hoped to come to a "global solution" on the issue "this year".
The banks are: Credit Suisse, Julius Bär, Wegelin, Banque cantonale de Zurich (ZKB), la Banque cantonale de Bale (BKB), Neue Zürcher Bank (NZB), HSBC, LLB, in addition to the Israeli banks Leumi, Hapoalim and Mizrahi.