British-based PFC Auctions says the blood sample was taken from Ronald Reagan after the failed 1981 assassination attempt against him.
The PFC website put the latest bid for the vial at £6,270 ($9,910) on Tuesday.
John Heubusch, executive director of the foundation, said that, if true, it would fight to stop “this craven act”.
“If indeed this story is true, it’s a craven act and we will use every legal means to stop its sale or purchase,” he said.
John Heubusch said the hospital where Ronald Reagan had been treated had assured the foundation that an inquiry was under way into “how something like this could possibly happen”.
PFC Auctions, based on Guernsey in the British Channel Islands, displayed a picture of the vial on its website bearing a label showing the president’s name.
The lot includes a letter of provenance from the seller who says their late mother worked at the laboratory which carried out blood testing for George Washington University Hospital after Mr Reagan was shot.
“These articles have actually been in my family’s possession since… the day that President Reagan was shot in Washington DC,” the letter reads.
Ronald Reagan suffered a punctured lung and internal bleeding when he was shot by John Hinckley Jr. outside the Washington Hilton Hotel.
John Hinckley Jr. was later found not guilty by reason of insanity and is being treated at a psychiatric hospital.
Ronald Reagan, who went on to serve two terms as president, died at the age of 93 in 2004.
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