It comes as US deaths from the pandemic have surged past 2,000 for two days in a row and President-elect Joe Biden warned of a “dark winter” trying to fight the virus.
The toll from Covid-19 reached its second-highest level ever on Wednesday with 2,811 lives lost, one short of the record from April 15.
Mr Bush's Chief of Staff Freddy Ford told CNN that the former Republican President would take the vaccine once it had been administered to those populations that need it most.
"First, the vaccines need to be deemed safe and administered to the priority populations", he said.
"Then, President Bush will get in line for his, and will gladly do so on camera."
Mr Clinton's press secretary then told CNN that the former Democrat President would also be happy to take the vaccine on camera.
Meanwhile, Barack Obama told SiriusXM that he trusted Anthony Fauci "completely" and that he "may end up taking it on TV.
"People like Anthony Fauci, who I know, and I've worked with, I trust completely", he said.
"So, if Anthony Fauci tells me this vaccine is safe, and can vaccinate, you know, immunise you from getting Covid, absolutely, I'm going to take it."
Nearly 200,000 new US cases were reported on Wednesday, with record hospitalisations approaching 100,000 patients.
The sobering data came as the head of the US Centers for Disease Control and Prevention on Wednesday warned that December, January and February were likely to be "the most difficult time in the public health history of this nation."
CDC Director Dr. Robert Redfield told an event hosted by the US Chamber of Commerce that the United States could start losing around 3,000 people - roughly the number that died in the attacks of Sept. 11, 2001 - each day over the next two months.
"The mortality concerns are real and I do think unfortunately before we see February, we could be close to 450,000 Americans that have died from this virus," Redfield said. The US death toll since the start of the pandemic stands at around 273,000.