Biden could also give Murthy an expanded title or elevate the role of the surgeon general within the administration, according to sources close to the transition, though no decision has been made.
Murthy is reportedly no longer under consideration to lead the Department of Health and Human Services.
Jeffrey Zients, a senior transition official who served as a senior economic adviser to Obama, is also expected to coordinate the Biden administration's coronavirus response from the White House, working with Murthy to lead the effort, according to sources close to the transition.
Additionally, Biden has asked Dr. Anthony Fauci, the nation's top infectious disease expert, to serve as his chief medical adviser.
"I asked him to stay on in the exact same role he's had for the past several presidents, and I asked him to be a chief medical adviser for me as well and be part of the COVID team," Biden said in a CNN interview on Thursday.
Biden is also considering David Kessler, who served as Food and Drug Administration commissioner under President Bill Clinton and has advised his campaign and transition, to lead the FDA, along with Joshua Scharfstein, who served as a deputy commissioner during Obama's first term, according to a source close to the transition.
ABC News' Molly Nagle and Anne Flaherty contributed to this report.