Democrats who face re-election in states Trump won in 2016 face pressure to support measures seen as bipartisan.
Azar would head the Health and Human Services Department, which oversees Medicare, Medicaid and Obamacare, as well as drug regulation.
Sen. Al Franken plans an announcement Thursday after new allegations and growing calls for his resignation.
Trump joked that he will get "unanimous support" from Democrats on the GOP tax proposal.
Doubts are growing that Republican senators can pass an Obamacare replacement on their own, and Democrats have seized on that.