• "There were multiple conversations with the U.S. government about taking more supply in the second quarter," Pfizer board member Dr. Scott Gottlieb told CNBC.
• The former FDA chief said he was optimistic that the company and the Trump administration would eventually strike a deal for additional doses of Pfizer's Covid vaccine.
• "We want to work with the U.S. government but this has been a challenging process," added Gottlieb.

The U.S. government turned down an offer to secure more doses of Pfizer's Covid-19 vaccine in November, company board member Dr. Scott Gottlieb told CNBC on Monday.

The Trump administration in July locked in a deal for 100 million doses if the vaccine proved safe and effective, with an option to purchase up to 500 million more. Pfizer has said it intends to ship 50 million vaccine doses this year, which would be enough to inoculate 25 million people because it requires two shots.

"There were multiple conversations with the U.S. government about taking more supply in the second quarter. The company wasn't taken up on that offer, as recently as November," Gottlieb said on "Squawk Box."

The vaccine from Pfizer and German partner BioNTech on Friday received emergency authorization from the Food and Drug Administration, and some Americans are likely to receive the first shots Monday.

Gottlieb, a former FDA chief under President Donald Trump, first indicated last week that the U.S. government had declined to exercise the option in its agreement with Pfizer for additional vaccine doses. He told CNBC that Pfizer offered the U.S. doses from its Michigan manufacturing plant for the second quarter of 2021. The company is already sending the U.S. all of its first-quarter supply from that plant, he said.

Gottlieb, a CNBC contributor, said last week that the company's overture to the government in November came after interim trial data was released showing the vaccine to be more than 90% effective in preventing Covid-19.

Pfizer has since entered into agreements with other countries for its second-quarter allotment of vaccine doses, Gottlieb said Monday. However, he expressed optimism that the company and the U.S. government will be able to strike a deal to acquire more doses.

"I think they're going to work this out. I think hopefully we'll find a way to increase supply and be able to get the government what the government needs," Gottlieb, who led the FDA from 2017 to 2019, said Monday. "This is an American company. We want to work with the U.S. government but this has been a challenging process because there have been multiple conversations happening as recently as November and now they're coming back and wanting to restart those conversations when other commitments have been made in the interim."

The Department of Health and Human Services pushed back against Gottlieb's initial remarks last week. In a statement posted to Twitter on Dec. 8, HHS said that Operation Warp Speed, the Trump administration's vaccine development initiative, never turned down "an offer from Pfizer for any number of millions of doses having a firm delivery date and quantity."

"We remain confident that across our portfolio of multiple vaccines we will have enough doses for any American who wants a vaccine by the end of Q2 2021," the HHS statement also said. The government has locked in vaccine doses from other companies, such as Moderna and Johnson & Johnson. The FDA is set to meet later this week to review Moderna's emergency use application, while J&J said last week it expects to be able to seek clearance from the agency for its vaccine in February.

Pfizer's chairman and CEO, Dr. Albert Bourla, echoed Gottlieb's optimism on a potential deal, telling CNBC he believes the company and the U.S. government should be able to settle on an agreement for additional vaccine doses. Bourla said the government is seeking an additional 100 million doses from Pfizer.

"We are in a position to provide them, but we are working the timeframe," Bourla said Monday, also on "Squawk Box." "We can provide a lot of that in the third quarter. The U.S. government wants it in the second quarter. We are working very collaboratively to try to find a solution and be able to allocate those 100 million in the second quarter if possible or a lot of them."

However, Bourla said, "We haven't signed an agreement [yet]."

HHS did not immediately respond to CNBC's request for comment on Gottlieb's remarks Monday.