The most-mentioned names include Diane Wood and Merrick Garland, both appeals-court judges, and Elena Kagan, the solicitor-general.
She met Mr Obama when he was a humble law lecturer, and is now his solicitor-general.
If appointed, she would have to recuse herself from a large number of cases in her first year on the Supreme Court, because as solicitor-general she represented the government in their early stages.
